Brief Information on the Surfaces Not Included in the Basic Content of the Encyclopedia

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Brief information on some classes of the surfaces which cylinders, cones and ortoid ruled surfaces with a constant were not picked out into the special section in the encyclo- distribution parameter possess this property. Other properties pedia is presented at the part “Surfaces”, where rather known of these surfaces are considered as well. groups of the surfaces are given. It is known, that the Plücker conoid carries two-para- At this section, the less known surfaces are noted. For metrical family of ellipses. The straight lines, perpendicular some reason or other, the authors could not look through to the planes of these ellipses and passing through their some primary sources and that is why these surfaces were centers, form the right congruence which is an algebraic not included in the basic contents of the encyclopedia. In the congruence of the4th order of the 2nd class. This congru- basis contents of the book, the authors did not include the ence attracted attention of D. Palman [8] who studied its surfaces that are very interesting with mathematical point of properties. Taking into account, that on the Plücker conoid, view but having pure cognitive interest and imagined with ∞2 of conic cross-sections are disposed, O. Bottema [9] difficultly in real engineering and architectural structures. examined the congruence of the normals to the planes of Non-orientable surfaces may be represented as kinematics these conic cross-sections passed through their centers and surfaces with ruled or curvilinear generatrixes and may be prescribed a number of the properties of a congruence of given on a picture. The surfaces suggested by L.S. Pontryagin straight lines, the order of which is equal to four. are studied in a paper [1] as a two-metric ruled manifolds. In the site [10], a ruled surface “Swallow Surface”, cyclic A generatrix of this surface is a straight line moving at the surfaces “Horn”, “Feder I”, «Frdre II» with generatrix circles space under corresponding law. Togino Kadzuto [2] finds the of the variable radius lying at the planes of pencil. conditions, under which conjugation of several Kuen sur- The variety of the self-intersecting surfaces such as faces (special algebraic surfaces) gives a C2-regular surface. “Twisted Fano”, “Twisted Plane” (Fig. 1), “Stiletto”, In a work [3], an equation of a Bress hyperboloid for a case of “Lemniscape”, “Slipper”, “Fano Planes”, “Pseudocatenoid”, general motion is obtained. The possible cases were inves- “Triaxial Teardrop”, there are shown in the site [11]. tigated and it was proved, that in the general case, geometric The exotic surfaces “Jeener’s klein surface”, “Bonan– locus of points with zero tangent accelerations is the one- Jeener’s klein surface 1” (Fig. 2), “Bent horns surface”, sheeted hyperboloid, but at a special case, these are cone, “Triangular trefoil”, “Triple corkscrew”, “Tori link” con- hyperbolic paraboloid, circular cylinder, hyperbolic cylinder, sisting of the repeating segments can be seen in the site [12]. two planes, straight line, or locus can be absent at all. The special Euler cones are mentioned in an article [4]. Surfaces with the constant equiaffine invariants were stud- ded by N.M. Onischuk [5]. This group of surfaces contains the second order surfaces and some more 6 surfaces. Bergmann Horst [6] considered inscribe and circum- scribeellipsoids of Steiner for n-metric simplex satisfying the demand of the extreme volume. They generalize the problem solved by Steiner for the plane case. In a paper of G. Brauner [7], ruled surfaces at the three- dimensional Euclidian space permitting the conform map- ping different from the isometry and similarity keeping their straight generatrixes are considered. It is proved, that only Fig. 1

S.N. Krivoshapko and V.N. Ivanov, Encyclopedia of Analytical Surfaces, 691 DOI 10.1007/978-3-319-11773-7, © Springer International Publishing Switzerland 2015 692 Brief Information on the Surfaces Not Included in the Basic Content of the Encyclopedia

Fig. 3

Fig. 2 References Several non-orientable surfaces of constant negative “ ” “ ” 1. Dmitrieva NP (1980) Graphic definition of the surface of Gaussian curvature ( Kink Surface , Kuen Surface , Pontryagin. Geometr Proektir Krivyh Liniy i Poverhnostey. “Breather Surface”, Two-Soliton and Three-Soliton Sur- Leningrad, 10-18, 2 refs, Ruk. Dep. v VINITI, Mar 31, 1980; faces), a number of algebraic surfaces given in implicit form No. 1299-80 Dep 2 (Cayley Cubic, Pretzel Surface, Pilz Surface, Orthocircles) 2. Togino K (1970) A surface of the class C formed from Kuen’s surfaces. J Mech Lab 24(1):16-22 (in Japan) one can see in the site [13]. 3. Iliev V (1970) Design of Bress hyperboloid. Nauchn Tr Vissh In-t Having appeared at the end of 70th years, the concepts Mashinostr, Mehaniz i Elektrifik Selsk Stop Ruse, 12(3):43-48 (in “fractal” and “fractal geometry” after the middle of the 80th Bulgarian) came into use of mathematicians and programmers. The 4. Fepl St (1971) Über spezielle Eulerkegeln. Matem vestnik 8 “ ” “ ” (4):363-366 (in German) word fractal comes from the Latin word fractus that 5. Onischuk NM (2005) Surfaces with the constant equiaffine means “fractional” and “frahgere” i.e. “break”. A fractal invariants. Izv Tomskogo Polytehnicheskogo Univ, 308(4):6-9 (4 surface is a surface consisting of the self-similar segments. refs) “Estestven. Nauki” Fractals may describe many physical phenomenon and nat- 6. Horst B (1983) Steinerellipsoide. Elem Math 38(6):137-142 7. Brauner H (1980) Die erzengendentreuen konformen Abbildungen ural formation such as mountains, clouds, trees, landscapes aus Regelflächen. Arch Math 33(5):470-477 with the good exactness. Firstly, the fractal nature of our 8. Palman D (1971) Über eine Strahlenkongruenz 4. Ordnung und 2. world was noticed by Benoît B. Mandelbrot. The main Klasse Glass mat 6(2):313-324 salient feature of the fractals is their continuous self-simi- 9. Bottema O (1971) Eine dem Plückerschen Konoid zugeordnete larity. Fractal surface consists of polygon or bi-polynomial Strahlenkongruenz. Glass mat 6(2):307-312 10. Parametrische Flächen und Körper. http://www.3d-meier.de/tut3/ surfaces given by chance. In the machine graphics, fractals 11. Bourke P (2007) Surfaces and Curves. University of Western are constructed by simple and quick iteration algorithms Australia, Australia. http://local.wasp.uwa.edu.au/*pbourke [14]. In Fig. 3 [15], an elliptic paraboloid designed with the 12. Mathematical Imagery by Jos Leys: http://www.josleys.com/show_ help of methods of fractal geometry is shown. In the ency- gallery.php?galid=274 13. Virtual Math Museum (3DXM Consortium) (2004-2006) http:// clopedia, these surfaces are not described. virtualmathmuseum.org/gallery4.html These are surfaces which are used usually for the 14. Muhin OI Computer Graphics: http://stratum.ac.ru/textbooks/ acquaintance with topologic objects and with theorems or they kgrafic/contents.html fi are used for the illustrations of topological works and ideas. 15. Vyzantiadou MA, Avdelas AV, Za ropoulos S (2007) The application of fractal geometry to the design of grid or reticulated Such surfaces as limaçon-shaped dunce hat, Duns egg, Cayley shell structures. Comput Aided Des 39(1):51-59 cubic, Whitney bottle, surfaces of Morin, Seifert, Haken and 16. Francis GK (1987–1988) A Topological Picturebook. Springer, some others are described in papers and books dealing with Berlin, p. 240 topology (see, for example, a book of Francis [16]). 17. Biswas I, Huisman J (2007) Rational real algebraic models of topological surfaces. Doc Math 12:549-567 Classification of All Surfaces Presented in the Encyclopedia

(class—subclass—group—subgroup—surfaces)

1. Ruled Surfaces

1.1. Ruled Surfaces of Zero Gaussian Curvature

1.1.1. Torse Surfaces (Torses) The literature on geometry and strength analysis of shells in the form of developable surfaces ■ Open (evolvent) helicoid ■ Monge ruled surface with the circular cylindrical directing surface ■ Developable conic helicoid ■ Developable helicoid with a cuspidal edge on the paraboloid of revolution ■ Parabolic torse ■ Torse with an edge of regression on the ellipsoid of revolution ■ Torse with an edge of regression on one sheet hyperboloid of revolution ■ Torse with an edge of regression given as x=e–t cost; y=e–t sint, z=e–t ■ Torse with an edge of regression given as x=v– v3/3; y=v2; z=a(v + v3/3) ■ Torse with an edge of regression in the form of a line of intersection of two cylinders with the perpendicular axes ■ Torse with an edge of regression in the form of hyperbolic helical line ■ Torse with a given line of curvature in the form of the second order parabola ■ Torse with generating straight lines lying in the normal planes of a spherical curve Developable helical surface with slope angles of straight generators from 0o till 90o (it is presented in the Class “9. Helix-shaped surfaces”) Ruled conic limaçon of revolution (it is presented in the Subclass “4.2. Monge surfaces with a conic directrix surface”) Developable surfaces with two plane directrix curves ■ Torse with two parabolas with intersecting axes ■ Torse with two parabolas lying in intersecting planes but with parallel axes ■ Torse with two ellipses lying in parallel planes and with parallel axes ■ Torse with two parabolas having one common axis and lying in intersecting planes ■ Torse with two parabolas of the second and forth order placed in parallel planes and with parallel axes ■ Torse with parabola and circle in parallel planes ■ Torse with parabola and ellipse in parallel planes ■ Torse with hyperbola and parabola in parallel planes ■ Torse with two ellipses placed in mutually perpendicular planes Torse with two parabolas lying in mutually perpendicular planes and with the apexes on the same coordinate axis (synonym is parabolic torse)

S.N. Krivoshapko and V.N. Ivanov, Encyclopedia of Analytical Surfaces, 693 DOI 10.1007/978-3-319-11773-7, © Springer International Publishing Switzerland 2015 694 Classification of All Surfaces Presented in the Encyclopedia

The third and fourth order developable surfaces ■ Torse with an edge of regression in the form of the line of intersection of circular cylinder and circular cone Parabolic torse (it is presented in the Group “1.1.1. Torse surfaces (torses)”) Surfaces of constant slope ■ Torse of constant slope with directrix parabola ■ Torse of constant slope with directrix catenary ■ Torse of constant slope with an edge of regression on one sheet hyperboloid of revolution ■ Torse of constant slope with directrix ellipse Torse of constant slope with directrix curve in the form of evolvent of the circle (synonyms are open (evolvent) helicoid and Monge’s ruled surface with circular cylindrical directing surface) Conical helicoid of constant slope (synonym is developable conic helicoids) Surface of constant slope with an edge of regression on a paraboloid of revolution (synonym is developable helicoid with a cuspidal edge on the paraboloid of revolution) Surface of constant slope with directrix circle (synonym is conical surface of revolution) Surface of constant slope with an edge of regression given as x=v–v3/3; y=v2; z=a(v + v3/3) (synonym is torse with an edge of regression given as x=v–v3/3; y=v2; z=a(v + v3/3)) Torse with an edge of regression on the ellipsoid of revolution (it is presented in the Group “1.1.1. Torse surfaces (torses)”

1.1.2. Cylindrical Surfaces ■ Cylindrical surfaces presented in the encyclopedia ■ Cylindrical helical strip ■ Elliptical cylinder ■ Parabolic cylinder ■ Hyperbolic cylinder ■ Right cylindrical surface with directrix semi-cubical parabola ■ Right cylinder with a directrix catenary ■ Right astroidal cylindrical surface ■ Evolvent cylindrical surface ■ Lemniscate cylinder ■ Right cylinder with a directrix logarithmic spiral ■ Right cylinder with a directrix hyperbolic spiral ■ Right cylinder with a directrix spiral of Fermat ■ Right cylinder with a directrix spiral of Archimedes ■ Surface of cylindrical ﬂexible hopper-type bin for the keeping of dry materials ■ Cylindrical surface “Eight” ■ Cylindrical-and-conical spiral strip ■ Oblique circular cylinder ■ Oblique elliptical cylinder ■ Oblique parabolic cylinder ■ Oblique hyperbolic cylinder Cylindrical surface of revolution (it is presented in the Class “35. The second order surfaces” Right wave-shaped cylindrical surface (it is presented in the Class “25. Wave-shaped, waving, and corrugated surfaces”) Elliptical cylinder with two directrix circlers in mutually perpendicular planes (synonym is cylindroid with two directrix circles lying in mutually perpendicular planes) Epicycloidal cylinder (it is presented in the Group “34.1.2. Rotational surfaces with axoids “cylinder–cylinder””) Hypocycloidal cylinder (it is presented in the Group “34.1.2. Rotational surfaces with axoids “cylinder–cylinder””) Cylindrical helical strip of variable pitch (it is presented in the Subclass “7.2. Helical surfaces of variable pitch”) Sinusoidal cylindrical surface (it is presented in the Class “25. Wave-shaped, waving, and corrugated surfaces”) Cylindrical-and spherical spiral-shaped strip (it is presented in the Class “9. Spiral-shaped surfaces”) Cylindrical helical strip with given slope angles of tangents at the beginning and at the end of generatrix cylindrical helical line of centers of variable pitch (it is presented in the Subclass “7.2. Helical surfaces of variable pitch”) Classification of All Surfaces Presented in the Encyclopedia 695

1.1.3. Conical Surfaces ■ Key conical surfaces presented in the encyclopedia The literature on geometry and strength analysis of shells in the form of conical surfaces ■ Elliptical conical surface ■ Oblique circular conical surface ■ Oblique elliptical conical surface ■ Conical surface with a directrix Agnesi curve ■ Conical surface with a directrix curve in the form of Cartesian folium ■ Evolvent conical surface ■ Hyperbolic conical surface ■ Right conical surface with a plane director curve in the form of circular sinusoid ■ Spiral conical strip ■ Conical surface with a directrix curve on a sphere ■ Helical cone Conical surface of revolution (it is presented in the Class “35. The second order surfaces”) Parabolic conical surface (it is presented in the Class “35. The second order surfaces”) Cubic cone (it is presented in the Subclass“36.1. Algebraic surfaces oh the third order”) Right conical sinusoidal wave-shaped surface (it is presented in the Class “25. Wave-shaped, waving, and corrugated surfaces”) Waving conical surface in lines of principle curvatures with inner vertex (it is presented in the Class “25. Wave- shaped, waving, and corrugated surfaces”) Rotational surface with axoids “cone–cone” generated by a straight line coming through the common vertex of the axoids (it is presented in the Group “34.1.3. Rotational surfaces with axoids “cone–cone””) Rotational surface with axoids “plane–cone” generated by a straight line coming through the vertex of a mobile cone (it is presented in the Group “34.1.4. Rotational surfaces with axoids “plane–cone»”) Ruled conic limaçon of revolution (it is presented in the Subclass “4.2. Monge surfaces with a conic directrix surface”) Honeycomb conical surface (it is presented in the Class “25. Wave-shaped, waving, and corrugated surfaces”)

1.2. Ruled Surfaces of Negative Gaussian Curvature

The literature on geometry and analysis of shells in the form of ruled surfaces of negative total curvature ■ Oblique helicoid ■ Ruled surface with straight generatrixes passing through a logarithmic spiral and intersecting the ﬁxed axis under constant angle ■ Spherical helicoid Right helicoid (it is presented in the Class “19. Minimal Surfaces”) Hyperboloid of one sheet (it is presented in the Class “35. Surfaces of the second order”) One-sheet hyperboloid of revolution (it is presented in the Class “2. Surfaces of revolution”) Convolute helicoid (it is presented in the Group “7.1.1. Ruled helical surfaces”) Helical surface generated by binormals of a cylindrical helix (it is presented in the Group “7.1.1. Ruled helical surfaces”) Ruled surface of the trajectory of movement of straight generatrix of an evolvent helicoid in the process of its parabolic bending (it is presented in the Class “8. Spiral surfaces”) Spiral surface with straight generatrixes in the planes of pencil (it is presented in the Class “8. Spiral surfaces”) Ruled rotational surfaces of negative total curvature (they are presented in the Subclass “34.1. Rotational surfaces”) Edlinger’s surfaces (they are presented in the Class “29. Edlinger’s surfaces”) Ruled spiroidal surfaces of negative Gaussian curvature (they are presented in the Subclass “34.2. Spiroidal Surfaces”) Spiroidal ruled surfaces of Rachkovskaya–Kharabaev (they are presented in the Group “34.2.5. Spiroidal ruled surfaces of Rachkovskaya – Kharabaev”) Helix-shaped preliminarily twisted strip (it is presented in the Class “10. Helix-shaped surfaces”) Ruled surface of Skidan (it is presented in the Class “25. Wave-shaped, waving, and corrugated surfaces”) 696 Classification of All Surfaces Presented in the Encyclopedia

1.2.1. Catalan Surfaces

■ Whitney umbrella (Cartan umbrella) ■ Pseudo-developable helicoid ■ Ruled rotor cylindroid ■ Saddle in the drum Hyperbolic paraboloid (it is presented in the Class “35. Surfaces of the second order”) Elliptic helicoid (it is presented in the Class “10. Helix-shaped surfaces”) Cayley surface (it is presented in the Class “3. Translation surfaces”) Cubic surface x3 +xy+z=0 (it is presented in the Subclass “36.1. The third order algebraic surfaces”) Spiral ruled surface with straight generatrix perpendicular to an axis of a directrix conic spiral and to the tangent of the same spiral (it is presented in the Class “8. Spiral surfaces”) Right helicoidal surface with variable pitch (it is presented in the Subclass “7.2. Helical surfaces of variable pitch”) Pseudo-developable helix-shaped surface with variable pitch (it is presented in the Class “10. Helix-shaped surfaces”) Right spherical helicoid (synonym is spherical helicoid) Right waving helicoid (it is presented in the Class “25. Wave-shaped, waving, corrugated surfaces”) Conoids Right helical conoid (synonym is right helicoid) ■ Parabolic conoid ■ Conoid with a directrix circle ■ Conoid with a directrix catenary ■ Right sinusoidal conoid ■ Right conoid with a directrix parabola the axis of which is parallel to the axis of conoid ■ Evolvent conoid ■ Plücker conoid ■ Wallis’s conical edge ■ Zindler’s conoid ■ Continuous topographic ruled surface with distributing ellipse Cylindroids ■ Cylindroid with two directrix ellipses ■ Cylindroid with two directrix circles lying in mutually perpendicular planes ■ Ball’s cylindroid ■ Cylindroid with a parabola and a sinusoid lying on the parallel ends ■ Frezier’s cylindroid

1.2.2. Twice Oblique Cylindroids ■ Twice oblique trochoid cylindroid

2. Surfaces of Revolution

■ Surfaces of revolution presented in the encyclopedia ■ One-sheet hyperboloid of revolution ■ Fairing of cycloidal type ■ Pseudo-sphere ■ Paraboloid of revolution ■ Circular torus ■ Elliptic torus ■ Surface of revolution of a curve z = bexp(–a2x2) around the z axis ■ Two-sheeted hyperboloid of revolution Classification of All Surfaces Presented in the Encyclopedia 697

■ Surface of conjugation of two coaxial cylinders of different diameters ■ Surface of revolution «Wellenkugel» ■ Surface of conjugation of coaxial cylinder and cone ■ Surface formed by rotation of a meridian in the form of semi-cubical parabola ■ Surface of revolution of a hyperbola z = b/x around the Oz axis ■ Parabolic humming-top ■ Surface of revolution of an astroid ■ Astroidal torus ■ Surface of revolution of the Agnesi curl ■ Deformed sphere ■ Surface of revolution of a parabola ■ Parabolic-and- logarithmic surface of revolution ■ Hyperbolic-and- logarithmic surface of revolution ■ Bullet nose ■ The fourth order paraboloid of revolution ■ Surface of revolution with damping circular waves ■ “Kiss surface” ■ Soucoupoid ■ Globoid (toroid) ■ Surface of revolution of a usual cycloid ■ Pseudo-catenoid ■ Surface of revolution «Pear» ■ Surface of revolution of a general sinusoid ■ Corrugated surface of revolution of a general sinusoid ■ Surface of revolution of a parabola of arbitrary position ■ Surface of revolution of a biquadrate parabola ■ Ellipsoid of revolution ■ Ding–Dong surface ■ “Eight surface” ■ Surface of revolution “Egg” of the fourth order ■ Surface of revolution “Egg” of the third order ■ Piriform Surface ■ “Drop” Kappa Surface (it is presented in the Class “6. Continuous topographic and topographic surfaces”) Spherical surface (sphere) (it is presented in the Subclass “23.1. Surfaces of constant positive Gaussian curvature”) Cylindrical surface of revolution (it is presented in the Class “35. The second order surfaces”) Conical surface of revolution (it is presented in the Class “35. The second order surfaces”) Catenoid (it is presented in the Class “19. Minimal surfaces”) Tractroid (antisphere, tractrisoid) (synonym is pseudosphere) Surface of revolution of an inclined sinusoid (synonym is helical surface with sinusoidal generatrix) Surface of revolution given by a harmonic function z = ln[x2 + y2]1/2 (it is presented in the Class “31. Surfaces given by harmonic functions “) Cycloidal torus (synonym is helical surface with generatrix cycloid) Surface of revolution of the evolvent of a circle (synonym is helical surface with the generatrix evolvent of a circle) Surface of revolution of a hyperbola of arbitrary position (it is presented in “Helical surface with generatrix curve in the form of hyperbola”) Surface of revolution of the Agnesi curl (it is presented in the Subclass “36.4. Algebraic surfaces of the sixth order”) Cyclic surfaces of revolution (it is presented in the Class “17. Cyclic surfaces”) The literature on analysis of shells in the form of surfaces of revolution 698 Classification of All Surfaces Presented in the Encyclopedia

2.1. Middle Surfaces of Bottoms of Shells of Revolution Made by Winding of One Family of Threads Along the Lines of Limit Deviation

2.2. Middle Surfaces of Bottoms of Shells of Revolution Made by Plane Winding of Threads

2.3. Middle Surface of Bottom of Shell of Revolution Made by Winding of Threads Along Geodesic Lines

2.4. Middle Surfaces of Shells of Revolution with Given Properties

■ Surfaces of revolution with geometrically optimal rise ■ Middle surface of non-bending shell of revolution under uniform pressure Surface of conjugation of coaxial cylinder and cone (it is presented in the Class “2. Surfaces of revolution”)

2.5. Surfaces of Revolution with Extreme Properties

Circular torus (it is presented in the Class “2. Surfaces of revolution”) ■ Surface of catenoidal type ■ “PenKa”

2.6. The Surfaces of Delaunay

Catenoid (it is presented in the Class “19. Minimal surfaces”) Spherical surface (sphere) (it is presented in the Subclass “23.1. Surfaces of the constant positive Gaussian curvature”) Cylindrical surface of revolution (it is presented in the Class “2. Surfaces of revolution”)

2.6.1 Nodoid and Unduloid Surfaces of Revolution ■ Nodoid surface connecting two circular cones

3. Translation Surfaces

■ Translation surfaces presented in the encyclopedia

3.1. Surfaces of Right Translation

Cylindrical surfaces (it is presented in “Cylindrical surfaces presented in the encyclopedia”) Paraboloid of revolution (it is presented in the Class “2. Surfaces of revolution”) Elliptic paraboloid (it is presented in the Class “35. The second order surfaces”) Hyperbolic paraboloid (it is presented in the Class “35. The second order surfaces”) Surface of translation of circle along sinusoid (it is presented in the Subclass “17.3. Cyclic surfaces with a plane of parallelism”) Classification of All Surfaces Presented in the Encyclopedia 699

Surface of translation of circle along ellipse (it is presented in the Subclass “17.3. Cyclic surfaces with a plane of parallelism”) The ﬁrst Scherk’s minimal surface (it is presented in the Class “19. Minimal surfaces”) ■ Circular surface of translation ■ Cayley surface ■ Surface of translation of catenary along catenary ■ Surface of translation of circle along parabola ■ Surface of translation of sinusoid along parabola ■ Elliptic surface of translation ■ Shoe surface ■ Surface of translation of parabola along hyperbola ■ Surface of translation of sinusoid along sinusoid ■ Cycloidal surface of translation The literature on geometry and analysis of shells in the form of parallel translation surfaces

3.2. Translation Surfaces with Congruent Generatrix and Directrix Curves

Paraboloid of revolution (it is presented in the Class “2. Surfaces of revolution”) Biquadratic surface of translation (synonym is circular surface of translation) Bicatenary surface of translation (synonym is surface of translation of catenary along catenary) Bi-elliptical surface of translation (synonym is elliptic surface of translation) Bicosine surface of translation (it is presented in the Class “5. Surfaces of congruent sections”) Bicycloidal surface of translation (synonym is cycloidal surface of translation) Bisemicubic surface of translation (it is presented in the Class “5. Surfaces of congruent sections”) Literature on geometry of translation surfaces with congruent generatrix and directrix curves (it is given in the Class “5. Surfaces of congruent cross sections”)

3.3. Surfaces of Oblique Translation

■ Volkov’s diagonal circular surface of translation ■ Oblique parabolic surface of translation Oblique cosine surface of translation (synonym is bicosine surface of translation)

3.4. Velaroidal Surfaces

■ Sinusoidal velaroid ■ Surface of velaroidal type on annulus plan with two families of sinusoids ■ Parabolic velaroid ■ Elliptical velaroid

4. Carved Surfaces

■ Carved surfaces presented in the encyclopedia Tubular surfaces (they are presented in the Subclass “17.2. Normal cyclic surfaces”) 700 Classification of All Surfaces Presented in the Encyclopedia

4.1 Monge Surfaces with a Circular Cylindrical Directrix Surface

Monge ruled surfaces with a circular cylindrical directing surface (they are presented in the Group “1.1.1. Torse surfaces (torses)”) Monge surface with a cylindrical directrix surface and a circular meridian (synonym is a tubular surface with a plane line of centers in the form of the evolvent of a circle) ■ Monge surface with a cylindrical directrix surface and a parabolic meridian ■ Monge surface with a cylindrical directrix surface and with a sinusoid as meridian ■ Monge surface with a cylindrical directrix surface and a hyperbolic meridian ■ Monge surface with a cylindrical directrix surface and meridian in the form of a cycloid ■ Monge surface with a cylindrical directrix surface and meridian in the form of catenary

4.2. Monge Surfaces with a Conic Directrix Surface

■ Ruled conic limaçon of revolution

4.3. Carved Surfaces of General Type

■ Carved sinusoidal surface ■ Carved surface with directrix cubic parabola and generating ellipse ■ Carved surface with directrix ellipse and generatrix parabola ■ Carved surface with directrix cycloid and generatrix ellipse ■ Carved surface with directrix sinusoid and generatrix cycloid ■ Carved surface with directrix sinusoid and generatrix parabola ■ Carved surface with directrix sinusoid and generatrix ellipse ■ Carved surface with directrix ellipse and generatrix sinusoid The literature on geometry and application of shells in the form of carved surfaces

5. Surfaces of Congruent Sections

Kinematical surfaces of general type (they are presented in the Class “34. Kinematical surfaces of general type”) Monge surfaces (they are presented in the Class “4. Carved surfaces”) Helical surfaces (they are presented in the Class “7. Helical surfaces”) Translation surfaces with congruent generatrix and directrix curves (they are presented in the Class “3. Translation surfaces) Tubular surfaces (they are presented in the Subclass “17.2. Normal cyclic surfaces”) Cyclic surfaces of revolution (they are presented in the Class “17. Cyclic surfaces”) Right circular surface on a cylinder (it is presented in the Subclass “17.3. Cyclic surfaces with a plane of parallelism”) Sinusoidal helicoid (it is presented in the Class “25. Wave-shaped, waving, and corrugated surfaces”) ■ Bicosine surface of translation ■ Bisemicubic surface of translation ■ Twisted surface with congruent ellipses in parallel planes The literature on geometry, application, and analysis of shells in the form of surfaces of congruent section Classification of All Surfaces Presented in the Encyclopedia 701

6. Continuous Topographic and Topographic Surfaces

Surfaces of revolution (they are presented in the Class “2. Surfaces of revolution”) Surfaces of right translation (they are presented in the Class “3. Translation surfaces”) Cyclic surfaces with a plane of parallelism (they are presented in the Class “17. Cyclic surfaces”) Catalan surfaces (they are presented in the Subclass “1.2. Ruled surfaces of negative Gaussian curvature”) Continuous topographic ruled surface with distributing ellipse (it is presented in the Group “1.2.1. Catalan surfaces”) ■ Topographic surface with the given elliptical cross sections ■ “Trash Can” ■ “Paper Bag” ■ Kappa surface ■ Continuous topographic surface of Cassini

6.1. Aerodynamic Surfaces Given by Algebraic Plane Curves

■ Aerodynamic surface given by a continuous framework of elliptical ribs ■ Aerodynamic surfaces given by a continuous framework of water-lines in the form of generalized Agnesi curls ■ The 6th order surface with Cartesian folium, ellipse, Cartesian folium lying in three principal coordinate sections ■ Surface of the 5th order with parabola, ellipse, parabola lying in three principal coordinate sections ■ Aerodynamic surfaces with a continuous framework of plane curves presented in the encyclopedia Aerodynamic surfaces with a continuous framework of plane curves (they are presented in the Class “36. Algebraic surfaces of the high orders”) 7. Helical Surfaces

7.1. Ordinary Helical Surfaces

■ Ordinary helical surfaces presented in the encyclopedia

7.1.1. Ruled Helical Surfaces Right helicoid (it is presented in the Class “19. Minimal surfaces”) Oblique helicoid (it is presented in the Subclass “1.2. Ruled surfaces of negative Gaussian curvature”) Open (evolvent) helicoid (it is presented in the Group “1.1.1. Torse surfaces (torses)”) Pseudo-open helicoid (it is presented in the Subclass “1.2. Ruled surfaces of negative Gaussian curvature”) Cylindrical helical strip (it is presented in the Group “1.1.2. Cylindrical surfaces”) Helical surface formed by the principle normals of helix (synonym is right helicoid) Helical surface formed by the tangents of helix (synonym is evolvent helicoid) ■ Convolute helicoid ■ Helical surface generated by binormals of a cylindrical helix

7.1.2. Circular Helical Surfaces ■ Circular helical surface with generatrix circles in the planes of pencil ■ Tubular helical surface ■ Right circular helical surface ■ Circular helical surface with generating circles lying in the osculating plane of a helix of centers of the circles “The Saint Elias surface” (it is presented in the Subclass “17.4. Cyclic surfaces with circles in planes of pencil”) Surface of a helical pole (it is presented in the Subclass “17.3. Cyclic surfaces with a plane of parallelism””) Helix-shaped preliminarily twisted surface of circular cross section (it is presented in the Subclass “10.1. Helix-shaped preliminarily twisted surfaces with plane generatrix curve” and in the Group “7.1.2. Tubular helical surface”) Helical twisted surface with circles in the planes of pencil (it is presented in the Subclass “10.2. Helix-shaped twisted surfaces with plane generating curves in the planes of pencil”) 702 Classification of All Surfaces Presented in the Encyclopedia

7.1.3. Ordinary Helical Surfaces with Arbitrary Plane Generatrix Curves ■ Dini’s helicoid ■ Helical surface with parabolic generatrix of arbitrary position ■ Helical surface with sinusoidal generatrix ■ Helical surface with generatrix ellipse ■ Helical sinusoidal strip ■ Helical surface with generatrix curve in the form of evolvent of the circle ■ Astroidal helicoid ■ Helical surface with generatrix cycloid ■ Helical surface with generatrix curve in the form of hyperbola

7.2. Helical Surfaces of Variable Pitch

■ Helical surfaces of variable pitch presented in the encyclopedia ■ Cyclic helical surface with given slope angles of tangents at the beginning and at the end of a cylindrical helical line of centers of variable pitch ■ Cylindrical helical strip of variable pitch ■ Right helicoidal surface with variable pitch ■ Cyclic helical surface with a line of centers of variable pitch ■ Cylindrical helical strip with given slope angles of tangents at the beginning and at the end of the directrix cylindrical helical line of centers of variable pitch The literature on application and analysis of shells in the form of helical surfaces

8. Spiral Surfaces

■ Spiral surfaces presented in the encyclopedia The literature on geometry, application, and analysis of shells in the form of spiral and spiral-shaped surfaces Cylindrical-and-conical spiral strip (it is presented in the Group “1.1.2. Cylindrical surfaces”) Spiral conical strip (it is presented in the Group “1.1.3. Conical Surfaces”) Developable conic helicoid (it is presented in the Group “1.1.1. Torse surfaces (torses)”) Torse with an edge of regression given as x=e-t cost; y=e-t sint, z=e-t (it is presented in the Group “1.1.1. Torse surfaces (torses)”) Right circular spiral surface (it is presented in the Subclass “17.3. Cyclic surfaces with a plane of parallelism”) Circular spiral surface with a generatrix circle of constant radius lying in planes of a pencil (it is presented in the Subclass “17.4. Cyclic surfaces with circles in planes of a pencil”) Tubular spiral surface (it is presented in the Group “17.2.1. Tubular surfaces”) Cyclic surface with generatrix circles in planes of a pencil and with a plane line of centers in the form of a logarithmic spiral (it is presented in the Subclass “17.4. Cyclic surfaces with circles in planes of a pencil”) Right cylinder with a directrix logarithmic spiral (it is presented in the Group “1.1.2. Cylindrical surfaces”) Ruled surface with straight generatrixes passing through a logarithmic spiral and intersecting the ﬁxed axis under constant angle (it is presented in the Subclass “1.2. Ruled surfaces of negative Gaussian curvature”) ■ Spiral ruled surface with straight generatrix perpendicular to an axis of a directrix conic spiral and to the tangent of the same spiral ■ Spiral surface with straight generatrixes in the planes of pencil ■ Spiral surface with parabolic generatrix of arbitrary position ■ Spiral surface with a generatrix ellipse ■ Spiral surface with a hyperbolic generatrix ■ Spiral surface with generatrix in the form of cycloid Classification of All Surfaces Presented in the Encyclopedia 703

■ Spiral surface with a sinusoidal generatrix ■ Spiral surface with generatrix in the form of the evolvent of a circle ■ Spiral surface with directrix logarithmic spiral and with parabolic generatrix

9. Spiral-Shaped Surfaces

■ Spiral-shaped surfaces presented in the encyclopedia The literature on geometry, the application, and analysis of shells in the form of spiral and spiral-shaped surfaces (the literature is presented in the Class “8. Spiral Surfaces”) Developable helicoid with a cuspidal edge on the paraboloid of revolution (it is presented in the Group “1.1.1. Torse surfaces (torses)”) Cyclic surface with a generating circle in the planes of pencil and with a plane line of centers in the form of spiral of Archimedes (it is presented in the Group “17.4.1. Cyclic surfaces with circles in the planes of pencil and with a plane center-to-center line”) Cyclic surface with generatrix circles of variable radius and with a plane line of centers constructed about a circular cylinder (it is presented in the Group “17.4.1. Cyclic surfaces with circles in the planes of pencil and with a plane center-to-center line) Cyclic surface in a cylinder (it is presented in the Class “10. Helix-Shaped Surfaces”) Right circular spiral-shaped surface with a generatrix circle of variable radius (it is presented in the Subclass “17.3. Cyclic surfaces with a plane of parallelism”) Spiral-shaped surface with generatrix sinusoids and with a directrix line of constant pitch on a circular cone (it is presented in the Class “25. Wave-shaped, waving, and corrugated surfaces”) Normal cyclic surface with generatrix circles of variable radius and with a plane center-to-center line in the form of a logarithmic spiral (it is presented in the Group “17.2.2. Normal cyclic surfaces with generatrix circles of variable radius”) Conical surface with a directrix curve on a sphere (in is presented in the Group “1.1.3. Conical surfaces”) Tubular surface on the sphere (it is presented in the Group “17.2.1. Tubular surfaces”) Surfaces with a spherical directrix curve (they are presented in the Class “21. Surfaces with a spherical directrix curve) ■ Tubular surface with a line of centers on one-sheet hyperboloid of revolution ■ Ruled surface of the trajectory of movement of straight generatrix of an evolvent helicoid in the process of its parabolic bending ■ Snail surface ■ Steinbach screw ■ Spiral-shaped surface with parabolic generatrixes and directrix line of constant pitch on a circular cone ■ Hyperbolic helicoid ■ “Cornucopia” ■ Cylindrical-and-spherical spiral-shaped strip ■ Transcendental surface with two families of plane isotropic coordinate lines ■ Spiral-shaped surface with elliptical generatrixes and with a line of centers of constant pitch on a circular cone ■ “Seashell”

9.1. Spiral-Shaped Cyclic Surfaces with Circles of Variable Radius in the Planes of Pencil

■ Spiral screw ■ Spiral-shaped surface “Shell without vertex” ■ Spiral-shaped surface “Shell with vertex” 704 Classification of All Surfaces Presented in the Encyclopedia

10. Helix-Shaped Surface

■ Helix-shaped surfaces presented in the encyclopedia Normal cyclic helix-shaped surface consisting of identical elements (it is presented in the Class “17. Cyclic Surfaces”) Hyperbolic helicoid (it is presented in the Class “9. Spiral-shaped surfaces”) Right waving helicoid (it is presented in the Class “25. Wave-shaped, waving, and corrugated surfaces) Tubular helix-shaped surface with a line of centers of variable pitch (it is presented in the Subclass “17.2. Normal Cyclic Surfaces”) Spiroidal surfaces with axoids “cylinder - cylinder” (they are presented in the Subclass “34.2. Spiroidal Surfaces”) ■ Elliptic helicoid ■ Helix-shaped twisted strip with straight generatrixes in the planes of pencil ■ Helix-shaped preliminarily twisted strip ■ Helix-shaped surface with variable elliptical cross section ■ Helix-shaped surface with generatrix circle of variable radius lying on a plane ■ Cyclic surface in a cylinder ■ Quasi-helical cyclic surface with boundary circles given in advance ■ Developable helix-shaped surface with slope angles of straight generators changing from 0o till 90o ■ Pseudo-developable helix-shaped surface with variable pitch ■ Rotational oblique helicoid

10.1. Helix-Shaped Preliminarily Twisted Surfaces with Plane Generatrix Curve

■ Helix-shaped preliminarily twisted surface of elliptical cross section ■ Helix-shaped preliminarily twisted surface of circular cross section

10.2. Helix-Shaped Twisted Surfaces with Plane Generating Curves in the Planes of Pencil

■ Helix-shaped twisted surface of elliptical cross section in the planes of pencil ■ Helical twisted surface with circles in the planes of pencil

11. Blutel Surfaces

Quadrics (they are presented in the Class “35. The Second Order Surfaces”) Dupin’s cyclides (they are presented in the Group “17.1.2. Dupin’s Cyclides”)

12. Veronese Surfaces

Steiner surfaces (synonyms are the Roman surface, cross cap, Steiner surfaces of the ﬁrst and second types)

13. Tzitzéica Surfaces

■ Tzitzéica surface of the second order with the centroaffine invariant I = –a2 ■ Tzitzéica surface of the third order with the centroaffine invariant I = 1/27 ■ Tzitzéica surface of the second order with the centroaffine invariant I = a2 ■ Tzitzéica surface of the third order with the centroaffine invariant I = –4/27 Classification of All Surfaces Presented in the Encyclopedia 705

14. Peterson Surfaces

Monge surfaces with a circular cylindrical directrix surface (they are presented in the Subclass “4.1. Monge Surfaces With a Circular Cylindrical Directrix Surface”) Surfaces of translation of the 2nd order curves (they are presented in the Class “3. Translation surfaces”) Surfaces of revolution of the 2nd order curves (they are presented in the Class “2. Surfaces of Revolution”) Minimal surfaces of Peterson (they are presented in the Class “19. Minimal surfaces”) ■ Bendings of a three-axial ellipsoid

15. Surfaces of Bézier

■ Bicubic surface of Bézier The literature on application and geometry of surfaces of Bézier

16. Quasi-Ellipsoidal Surfaces

■ Quasi-ellipsoidal surface with three values of semi-axes

16.1. Quasi-Ellipsoidal Surfaces with Six Values of Semi-Axes

■ Quasi-ellipsoidal surface with concave segments between ribs ■ Ruled quasi-ellipsoidal surface ■ Quasi-ellipsoidal surface with convex segments between ribs

16.2. Quasi-Ellipsoidal Surfaces with Cylindrical Insertions

■ Quasi-ellipsoidal surface with convex segments and a cylindrical insertion

17. Cyclic Surfaces

■ Classiﬁcation of cyclic surfaces Literature on geometry and analysis of shells in the form of cyclic surfaces Circular helical surface with generating circles lying in the osculating plane of a helix of centers of the circles (it is presented in the Group “7.1.2. Circular helical surfaces”) Regular cyclic rotational surfaces (they are presented in “Cyclic rotational surface with axoids “cylinder–plane””)

17.1. Canal Surfaces

Tubular surfaces (they are presented in the Subclass “17.2. Normal cyclic surfaces”) 706 Classification of All Surfaces Presented in the Encyclopedia

17.1.1. Canal Surfaces of Joachimsthal Virich’s cyclic surface (it is presented in the Class “32. Surfaces of Joachimsthal”) Canal surfaces of Joachimsthal in the lines of principle curvature ■ Epitrochoidal surface ■ Joachimsthal cosine canal surfaces of the 1st type ■ Joachimsthal cosine canal surfaces of the 2nd type ■ Joachimsthal cosine canal surfaces of the 3d type

17.1.2. Dupin Cyclides ■ Dupin cyclides of the ﬁrst type (of the forth order) ■ Dupin cyclide of the second type (of the third order) Circular torus is Dupin cyclide of the third type but of the forth order (it is presented in the Class “2. Surfaces of revolution”)

17.2. Normal Cyclic Surfaces

17.2.1. Tubular Surfaces ■ Tubular surfaces presented in the encyclopedia Tubular helical surface (it is presented in the Group “7.1.2. Circular Helical Surfaces”) Circular torus (it is presented in the Class “2. Surfaces of revolution”) Wave-shaped torus on the sphere (it is presented in the Class “25. Wave-shaped, waving, and corrugated surfaces”) Helix-shaped preliminarily twisted surface of circular cross section (it is presented in the Class “10. Helix-shaped surfaces”) Tubular loxodrome (it is presented in the Class “21. Surfaces with spherical director curve”) Tubular surface with a center-to-center line on the one- sheet hyperboloid of revolution (it is presented in the Class “9. Spiral-shaped surfaces”) Tubular surface winding the sphere (it is presented in the Class “21. Surfaces with spherical director curve”) ■ Tubular spiral surface ■ Tubular surface with a plane line of centers in the form of the evolvent of a circle ■ Tubular surface with a plane line of centers in the form of a cycloid ■ Tubular surface with a plane parabolic line of centers ■ Tubular surface with a plane hyperbolic centerline ■ Tubular surface with a plane elliptical line of centers ■ Tubular surface with a plane sinusoidal centerline ■ Tubular surface on the sphere ■ Tubular helix-shaped surface with a center-to-center line of variable lead

17.2.2. Normal Cyclic Surfaces with Generatrix Circle of Variable Radius ■ Normal cyclic surfaces with generatrix circle of variable radius presented in the encyclopedia ■ Normal cyclic surface with plane circular line of centers and with a generator circle of variable radius ■ Normal cyclic helix-shaped surface consisting of identical elements ■ Normal cyclic surface with generatrix circles of variable radius and with a plane center-to-center line in the form of a logarithmic spiral ■ Normal cyclic surface with generatrix circle of variable radius and with a plane center-to-center line in the form of a conical spiral ■ Connecting canal for two cylindrical surfaces with parallel axes ■ Normal cyclic surface with an elliptical line of centers and with a generatrix circle of variable radius (the ﬁrst type) ■ Normal cyclic surface with an elliptical line of centers and with a generatrix circle of variable radius (the second type) Classification of All Surfaces Presented in the Encyclopedia 707

17.3. Cyclic Surfaces with a Plane of Parallelism

Right circular helical surface (it is presented in the Group “7.1.2. Circular helical surface”) Oblique circular conical surface (it is presented in the Group “1.1.3. Conical surface”) Oblique circular cylinder (it is presented in the Group “1.1.2. Cylindrical surfaces”) Circular surface of translation (it is presented in the Subclass “3.1. Surfaces of right translation”) Surface of translation of circle along parabola (it is presented in the Subclass “3.1. Surfaces of right translation”) Volkov’s diagonal circular surface of translation (it is presented in the Subclass “3.3. Surfaces of oblique translation”) ■ Right circular spiral surface ■ Surface of translation of a circle along a sinusoid ■ Surface of translation of a circle along an elliptical centerline ■ Surface of a helical pole ■ Right circular spiral-shaped surface with a generatrix circle of variable radius… ■ Right circular surface on a cylinder

17.4. Cyclic Surfaces with Circles in Planes of Pencil

■ Cyclic surfaces with circles in planes of pencil presented in the encyclopedia Circular helical surface with generatrix circles in the planes of pencil (it is presented in the Group “7.1.2. Circular Helical Surfaces”) Spiral-shaped cyclic surfaces with circles of variable radius in the planes of pencil (they are presented in the Class “9. Spiral-Shaped Surfaces”) Helix-shaped surface with generatrix circle of variable radius lying on a plane (it is presented in the Class “10. Helix- Shaped Surfaces”) Helical twisted surface with circles in the planes of pencil (it is presented in the Class “10. Helix-Shaped Surfaces”) Wave-shaped torus (synonym is cyclic surface with circles in the planes of pencil and with a waving line of centers on a cylinder) Cyclic surface in a cylinder (it is presented in the Class “10. Helix-shaped surfaces”) Quasi-helical cyclic surface with boundary circles given in advance (it is presented in the Class “10. Helix-shaped surfaces”) Cyclic helical surface with a line of centers of variable pitch (it is presented in the Subclass “7.2. Helical surfaces of variable pitch”) Cyclic helical surface with given slope angles of tangents at the beginning and at the end of a cylindrical helical line of centers of variable pitch (it is presented in the Subclass “7.2. Helical surfaces of variable pitch”) ■ Circular spiral surface with a generatrix circle of constant radius lying in planes of a pencil ■ “The Saint Elias surface” ■ Cyclic surface with circles in the planes of pencil and with a waving line of centers on a cylinder ■ Cyclic surface with circles in the planes of meridians of the sphere and with a center-to-center line on the same sphere ■ “Clover knot” ■ “Hornlet”

17.4.1. Cyclic Surfaces with Circles in the Planes of Pencil and with a Plane Center-to-Center Line Circular torus (it is presented in the Class “2. Surfaces of revolution”) Preliminarily twisted circular torus (it is presented in “Helix-shaped preliminarily twisted surface of circular cross section) Epitrochoidal surface (it is presented in the Group “17.1.1. Canal surfaces of Joachimsthal”) Dupin cyclides (they are presented in the Group “17.1.2. Dupin cyclides”) Canal surfaces of Joachimsthal (they are presented in the Group “17.1.1. Canal surfaces of Joachimsthal”) “Cornucopia” (it is presented in the Class “9. Spiral-shaped surfaces”) 708 Classification of All Surfaces Presented in the Encyclopedia

■ Cyclic surface with a generatrix circle in the planes of pencil and with a plane line of centers in the form of a logarithmic spiral ■ Cyclic surface with generatrix circles of variable radius and with a plane line of centers constructed about a circular cylinder ■ Cyclic surface with a generating circle in the planes of pencil and with a plane line of centers in the form of spiral of Archimedes ■ Cyclic surface with a generatrix circle in the planes of pencil and with a plane centerline in the form of the 2nd order curve ■ Cyclic surface with circles in planes of pencil and with a straight line of centers ■ Cyclic surface with circles in planes of pencil with a straight directrix and a ﬁxed straight of pencil that are lying on different sides of a plane center-to-center line ■ Cyclic surface with circles in planes of pencil, with a straight directrix and a ﬁxed straight of pencil that are lying on the same side of a plane center-to-center line ■ Cyclic surface with circles of variable radius in planes of pencil and with three straight parallel directrixes

17.5. Cyclic Surfaces of Revolution

Circular torus (it is presented in the Class “2. Surfaces of revolution”) Spherical surface (it is presented in the Class “23. Surfaces of the constant positive Gaussian curvature”) ■ Cyclic surface of revolution with the rotation axis parallel to the planes with generatrix circles ■ Cyclic surfaces of revolution with the rotation axis intersecting the planes with generatrix circles at the constant angle

18. One-Sided Surfaces

Steiner surface of the ﬁrst type (it is presented in “The Roman surface”) The literature on geometry of one-sided surfaces ■ One-sided ruled surface (Möbius strip) ■ Cross cap ■ The Roman surface ■ The Klein surface (The Klein bottle) ■ The Boy surface ■ The lemniscate one-sided surface

19. Minimal Surfaces

The literature on geometry of minimal surfaces and analysis of shells having the form of these surfaces ■ Catenoid ■ Right helicoid ■ Scherk’s minimal surface (the ﬁrst one) ■ Enneper’s surface ■ Schwarz surface ■ Neovius’ surface ■ Catalan’s surface ■ Bour’s minimal surface ■ Costa minimal surface ■ Gyroid ■ Henneberg minimal surface Classification of All Surfaces Presented in the Encyclopedia 709

■ Trinoid ■ Lichtenfels minimal surface ■ The second Scherk’s minimal surface ■ Richmond’s minimal surface Hoffman’s minimal surface (it is presented at the end of “Minimal surfaces presented in Internet sites”) Transcendental afﬁne minimal surface (it is presented in the Class “20. Afﬁne minimal surfaces”) Chen–Gackstatter surfaces (they are presented at the end of “Minimal surfaces presented in Internet sites”) Lopez Minimal Surface (it is presented at the end of “Minimal surfaces presented in Internet sites”) Oliveira’s minimal surface (it is presented at the end of “Minimal surfaces presented in Internet sites”)

19.1. Minimal Surfaces Pulled Over a Rigid Support Contour and Given by Point Frame

19.2. Minimal Surfaces with Free Boundaries

■ Minimal surfaces presented in Internet sites

19.3. Complete Minimal Surfaces

Catenoid (it is presented in the Class “19. Minimal surfaces”) Enneper’s surface (it is presented in the Class “19. Minimal surfaces”) Right helicoid (it is presented in the Class “19. Minimal surfaces”) Scherk’s minimal surface (it is presented in the Class “19. Minimal surfaces”) Richmond’s minimal surface (it is presented in the Class “19. Minimal surfaces”)

19.4. Minimal Surfaces of Peterson

Catenoid (it is presented in the Class “19. Minimal surfaces”)

19.5. Minimal Surfaces of Thomsen

Enneper’s surface (it is presented in the Class “19. Minimal surfaces”) ■ Minimal surface of Thomsen permitting transition to Enneper’s surface

19.6. The Chen-Gackstatter Surfaces

19.7. Algebraic Minimal Surfaces

Enneper’s surface (it is presented in the Class “19. Minimal surfaces”)

19.8. Embedded Triply-Periodic Minimal Surfaces

■ Several examples of embedded triply-periodic minimal surfaces 710 Classification of All Surfaces Presented in the Encyclopedia

20. Affine Minimal Surfaces

Enneper’s surface (it is presented in the Class “19. Minimal surfaces”) Elliptical paraboloid (it is presented in the Class “35. The 2nd order surfaces”) Minimal surfaces of Thomsen (they are presented in the subclass “19.5. Minimal surfaces of Thomsen”) Ttranscendental surface with two families of plane isotropic coordinate lines (it is presented in the Class “9. Spiral- shaped surfaces”) ■ Transcendental afﬁne minimal surface

21. Surfaces with Spherical Director Curve

■ Surfaces with a spherical director curve presented in the encyclopedia Spherical helicoid (it is presented in the Subclass “1.2. Ruled surfaces of negative Gaussian curvature”) Right spherical helicoid (synonym is spherical helicoid) Conical surface with a directrix curve on a sphere (it is presented in the Group “1.1.3. Conical surfaces”) Cylindrical-and-spherical spiral-shaped strip (it is presented in the Class “9. Spiral-shaped surfaces”) Tubular surface on the sphere (it is presented in the Group “17.2.1. Tubular surfaces”) Cyclic surface with circles in the planes of meridians of the sphere and with a center-to-center line on the same sphere (it is presented in the Subclass “17.4. Cyclic surfaces with circles in planes of pencil”) Wave-shaped torus on the sphere (it is presented in the Class “25. Wave-shaped, waving, and corrugated surfaces”) Torse with generating straight lines lying in the normal planes of a spherical curve (it is presented in the Group “1.1.1. Torse surfaces (torses)”) ■ Surface with a spherical director curve and with a parabolic generatrix in the planes of meridians of the sphere ■ Surface with a spherical director curve and with an elliptical generatrix in the planes of meridians of the sphere ■ Tubular loxodrome ■ Tubular surface winding the sphere

22. Weingarten Surfaces

Surfaces of revolution (they are presented in the Class “2. Surfaces of revolution”) Surfaces of the constant Gaussian curvature (they are presented in the Class “23. Surfaces of the constant Gaussian curvature”) Surfaces of the constant mean curvature (they are presented in the Class “24. Surfaces of the constant mean curvature”)

23. Surfaces of the Constant Gaussian Curvature

23.1. Surfaces of the Constant Positive Gaussian Curvature

■ Spherical surface (sphere) ■ Rembs’ surface ■ Sievert’s surface

23.2. Surfaces of the Constant Negative Gaussian Curvature

Pseudosphere (it is presented in the Class “2. Surfaces of revolution”) Dini’s helicoid (it is presented in the Group “7.1.3. Ordinary helical surfaces with arbitrary plane generatrix curves”) ■ Kuen surface The literature on geometry and analysis of shells in the form of surfaces of the constant negative Gaussian curvature Classification of All Surfaces Presented in the Encyclopedia 711

Surfaces of the constant zero Gaussian curvature Ruled surfaces of zero Gaussian curvature (they are presented in the Class “1. Ruled surfaces”)

24. Surfaces of the Constant Mean Curvature

Spherical surface (sphere) (it is presented in the Class “23. Surfaces of the constant positive Gaussian curvature”) Cylindrical surface of revolution (it is presented in the Class “35. The second order surfaces”) Minimal surfaces (they are presented in the Class “19. Minimal surfaces”) Nodoid and unduloid surfaces of revolution (they are presented in the Group “2.6.1. Nodoid and unduloid surfaces of revolution”) The literature on geometry of surfaces of the constant mean curvature

25. Wave-Shaped, Waving, and Corrugated Surfaces

The literature on geometry, analysis, and the application of shells in the form of wave-shaped, waving, and corrugated surfaces ■ Wave-shaped, waving, and corrugated surfaces presented in the encyclopedia… Right conical surface with a plane director curve in the form of circular sinusoid (it is presented in the Group “1.1.3. Conical surfaces”) Right sinusoidal conoid (it is presented in the Group “1.2.1. Catalan Surfaces”) Surface of translation of sinusoid along parabola (it is presented in the Class “3.Translation surfaces”) Surface of translation of sinusoid along sinusoid (it is presented in the Class “3. Translation surfaces”) Surface of revolution of a general sinusoid (it is presented in the Class “2. Surfaces of revolution”) Corrugated surface of revolution of a general sinusoid (it is presented in the Class “2. Surfaces of revolution”) Carved sinusoidal surface (it is presented in the Subclass “4.3. Carved surfaces of general type”) Carved surface with directrix sinusoid and generatrix ellipse (it is presented in the Subclass “4.3. Carved surfaces of general type”)) Carved surface with directrix ellipse and generatrix sinusoid (it is presented in the Subclass “4.3. Carved surfaces of general type”) Carved surface with directrix sinusoid and generatrix cycloid (it is presented in the Subclass “4.3. Carved surfaces of general type”) Cyclic surface with circles in the planes of pencil and with a waving line of centers on a cylinder (it is presented in the Subclass “17.4. Cyclic surfaces with circles in planes of pencil”) Spiral surface with a sinusoidal generatrix (it is presented in the Class “8. Spiral surfaces”) Wave-shaped torus (synonym is cyclic surface with circles in the planes of pencil and with a waving line of centers on the cylinder) Right circular surface on a cylinder (it is presented in the Subclass “17.4. Cyclic surfaces with a plane of parallelism“) Spiroidal surfaces with axoides “cylinder–cylinder” (they are presented in the Subclass “34.2. Spiroidal surfaces”) ■ Sinusoidal helicoid ■ Waving sinusoidal velaroid ■ Right conical sinusoidal wave-shaped surface ■ Waving conical surface in lines of principle curvatures with inner vertex ■ Honeycomb conical surface ■ Right wave-shaped cylindrical surface ■ Skidan’s ruled surface ■ Right waving helicoid 712 Classification of All Surfaces Presented in the Encyclopedia

■ Wave-shaped torus on the sphere ■ Sinusoidal cylindrical surface ■ Spiral-shaped surface with generatrix sinusoids and with a directrix line of constant pitch on a circular cone ■ Waving surface with the pseudo Agnesi curls of cylindrical type ■ Waving surface with the pseudo Agnesi curls on a circular plan ■ Corrugated paraboloid of revolution ■ Corrugated sphere ■ Ruled rotational surface with axoids “plane–cylinder” ■ Parabolic rotational surface with axoids “plane–cylinder” ■ Sphere with cycloidal crimps ■ Waving surface with cubical parabolas ■ Waving surface with semi-cubical parabolas

25.1. Waving Chains with Elliptical Cross Sections Limited by Surfaces of the 2nd Order

■ Waving chain with elliptical cross sections limited by an elliptical cylinder ■ Waving chain with elliptical cross sections limited by an elliptical cone ■ Waving chain with elliptical cross sections limited by an elliptical paraboloid

26. Surfaces of Umbrella Type

■ Surfaces of umbrella type presented in the encyclopedia The literature on geometry, the application, and analysis of umbrella shells and shells of umbrella type ■ Surface of umbrella type with parabolic generatrixes and with the opening at the vertex ■ Surface formed by parabolas but with radial waves damping in the central point ■ Waving ellipsoidal surface ■ Paraboloid of revolution with radial waves ■ Surface of umbrella type with parabolic generatrixes and with a circular opening at the vertex Skidan’s ruled surface (it is presented in the Class “25. Wave-shaped, waving, and corrugated surfaces”) Corrugated paraboloid of revolution (it is presented in the Class “25. Wave-shaped, waving, and corrugated surfaces”) Waving surface with the pseudo Agnesi curls on a circular plan (it is presented in the Class “25. Wave-shaped, waving, and corrugated surfaces”) Waving surface with cubical parabolas (it is presented in the Class “25. Wave-shaped, waving, and corrugated surfaces”) Sphere with cycloidal crimps (it is presented in the Class “25. Wave-shaped, waving, and corrugated surfaces”)

26.1. Surfaces of Umbrella Type with the Central Plane Point

■ Surface formed by cubic parabolas but with radial waves damping in the central point ■ Crossed trough ■ Umbrella surface formed by biquadratic parabolas and with astroidal level line ■ Surface of umbrella type formed by cubic parabolas on a cycloidal plan

26.2. Surfaces of Umbrella Type with Singular Central Point

■ Surface of umbrella type with a sinusoidal generatrix ■ Surface of umbrella type formed by semi-cubic parabolas on a cycloidal plan Classification of All Surfaces Presented in the Encyclopedia 713

27. Special Profiles of Cylindrical Products

■ Triangular profile of cylindrical fragment of a shaft for the profile detachable joint ■ Superellipses ■ Two types of aerodynamic cylindrical profiles ■ Generalized superellipses ■ Compound profiles formed by a curve and its mirror reflection ■ Profiles of products of trochoidal rotary machines

28. Bonnet Surfaces

Cylindrical surface of revolution (it is presented in the Class “35. The second order surfaces”) Right cylinder with a directrix logarithmic spiral (it is presented in the Group “1.1.2. Cylindrical surfaces”)

29. Edlinger’s Surfaces

One-sheet hyperboloid of revolution (it is presented in the Class “2. Surfaces of revolution”)

30. Coons Surfaces

■ Coons surfaces on a curvilinear quadrangular plane

31. Surfaces Given by Harmonic Functions

Right helicoid (z=carctg(y/x)) (it is presented in the Subclass “1.2. Ruled surfaces of negative Gaussian curvature”) ■ Surface of revolution given by the harmonic function z=ln[x2 + y2]1/2 ■ Harmonic surface of right translation of a sinusoid with changing amplitude ■ Harmonic surface of right translation

32. Surfaces of Joachimsthal

Canal surfaces of Joachimsthal (they are presented in the Group “17.1.1. Canal surfaces of Joachimsthal”) Surfaces of revolution (they are presented in the Class “2. Surfaces of revolution”) ■ Virich cyclic surface The literature on geometry of Joachimsthal surfaces

33. Saddle Surfaces

Saddle in the drum (it is presented in the Group “1.2.1. Catalan surfaces”) Ruled saddle surfaces (synonym is ruled surfaces of negative Gaussian curvature) Surfaces of the constant negative Gaussian curvature (they are presented in the Subclass “23.2. Surfaces of the constant negative Gaussian curvature”) Minimal surfaces (they are presented in the Class “19. Minimal surfaces “) 714 Classification of All Surfaces Presented in the Encyclopedia

■ Saddle surface of the ε class ■ Peano saddle ■ Narrowing saddle surface of Rosendorn ■ Flat saddle in the drum ■ Monkey saddle ■ Saddle surface with zero rotation of horn

34. Kinematical Surfaces of General Type.

The literature on geometry of rotational and spiroidal surfaces Surfaces of right translation (they are presented in the Class “3. Translation surfaces”)

34.1. Rotational Surfaces

Surfaces of revolution (they are presented in the Class “2. Surfaces of Revolution”) Oblique helicoid (it is presented in the Subclass “1.2. Ruled surfaces of negative Gaussian curvature”) Convolute helicoid (it is presented in the Group “7.1.1. Ruled helical surfaces”) Rotational surfaces with axoids “plane–cylinder” (they are presented in the Class “25. Wave-shaped, waving, and corrugated surfaces”) Monge surfaces with a circular cylindrical directrix surface (they are given in the Class “4. Carved surfaces”) Rotational surfaces with axoids “cone–plane» (synonym is Monge surface with a conic directrix surface)

34.1.1. Rotational Surfaces with Axoids “Cylinder–Plane” ■ Ruled rotational surface with axoids “cylinder–plane” ■ Cyclic rotational surface with axoids “cylinder–plane”

34.1.2. Rotational Surfaces with Axoids “Cylinder–Cylinder” ■ Ruled rotational surface of Lusta ■ Epicycloidal cylinder ■ Hypocycloidal cylinder ■ Rotational surface with axoids “cylinder–cylinder” formed by a straight not intersecting the axis of a mobile cylinder in the process of external rolling ■ Rotational surface with axoids “cylinder–cylinder” formed by a straight not intersecting the axis of a mobile cylinder in the process of internal rolling ■ Rotational surface with axoids “cylinder–cylinder” formed by a straight intersecting the axis of a mobile cylinder in the process of external rolling ■ Rotational surface with axoids “cylinder–cylinder” formed by a straight intersecting the axis of a mobile cylinder in the process of internal rolling

34.1.3. Rotational Surfaces with Axoids “Cone–Cone” ■ Rotational surface with axoids “cone–cone” generated by a straight line coming through the common vertex of the axoids (external rolling) ■ Rotational surface with axoids “cone–cone” generated by a straight line coming through the common vertex of the axoids (internal rolling) ■ Rotational e surface with axoids “cone–cone” formed by a straight intersecting the axis of a mobile cone in the process of external rolling ■ Rotational surface with axoids “cone–cone” formed by a straight parallel to the axis of a mobile cone in the process of external rolling ■ Rotational surface with axoids “cone–cone” formed by a parabola in the process of external rolling (the ﬁrst type) ■ Rotational surface with axoids “cone–cone” formed by a parabola in the process of external rolling (the second type) Classification of All Surfaces Presented in the Encyclopedia 715

34.1.4. Rotational Surfaces with Axoids “Plane–Cone” ■ Rotational surface with axoids “plane - cone” generated by a straight line coming through the vertex of a mobile cone ■ Rotational surface with axoids “plane–cone” formed by a straight intersecting the axis of a mobile cone ■ Rotational surface with axoids “plane–cone” formed by a straight parallel to the axis of a mobile cone

34.2. Spiroidal Surfaces

Ordinary helical surfaces (they are presented in the Subclass “7. 1. Ordinary helical surfaces”)

34.2.1. Spiroidal Surfaces with Axoids “Cylinder–Plane” ■ Regular ruled cylindrical spiroidal surface ■ Regular cyclic cylindrical spiroidal surface ■ Cylindrical helical limaçon with parabolic generatrix

34.2.2. Spiroidal Surfaces with Axoids “Cylinder–Cylinder” ■ Spiroidal surfaces with axoids “cylinder–cylinder” formed by a straight not intersecting the axis of a mobile cylinder in the process of external rolling ■ Spiroidal surface with axoids “cylinder–cylinder” formed by a straight not intersecting the axis of a mobile cylinder in the process of internal rolling ■ Spiroidal surface with axoids “cylinder–cylinder” formed by a straight intersecting the axis of a mobile cylinder in the process of external rolling ■ Spiroidal surface with axoids “cylinder–cylinder” formed by a straight intersecting the axis of a mobile cylinder in the process of internal rolling

34.2.3. Spiroidal Surfaces with Axoids “Plane–Cone” ■ Spiroidal surface with axoids “plane–cone” formed by a straight passing through the vertex of a mobile cone…

34.2.4. Spiroidal Surfaces with Axoids “Plane–Cylinder” ■ Spiroidal surface with axoids “plane–cylinder» formed by a straight parallel to the axis of a rolling cylinder ■ Spiroidal surface with axoids “plane–cylinder» formed by a straight intersecting the axis of a rolling cylinder ■ Parabolic spiroidal surface with axoids “plane–cylinder”

34.2.5. Spiroidal Ruled Surfaces of Rachkovskaya–Kharabaev ■ Spiroidal ruled surface of Rachkovskaya–Kharabaev with axoids “evolvent helicoid–right circular cone” ■ Spiroidal ruled surface of Rachkovskaya–Kharabaev with axoids “developable conical helicoid–right circular cone” ■ Spiroidal ruled surface of Rachkovskaya–Kharabaev with axoids “evolvent helicoid–right elliptical cone” ■ Kinematic ruled surface on the base of one-sheet hyperboloidal surfaces of revolution as ﬁxed and loose axoids (one axoid is located outside another) ■ Kinematic ruled surface on the base of one-sheet hyperboloidal surfaces of revolution as ﬁxed and loose axoids (one axoid is located in the interior of another)

35. The Second Order Surfaces

■ Quadratic surfaces presented in the encyclopedia ■ Hyperbolic paraboloid ■ Parabolic conic surface ■ Cylindrical surface of revolution ■ Conical surface of revolution ■ One-sheet hyperboloid ■ Ellipsoid 716 Classification of All Surfaces Presented in the Encyclopedia

■ Elliptic paraboloid ■ Hyperboloid of two sheets Spherical surface (sphere) (it is presented in the Subclass “23.1. Surfaces of the constant positive Gaussian curvature”) Two-sheeted hyperboloid of revolution (it is presented in the Class “2. Surfaces of Revolution”) Paraboloid of revolution (it is presented in the Class “2. Surfaces of Revolution”) Elliptical conical surface (it is presented in the Group “1.1.3. Conical surfaces”) Oblique circular conic surface (it is presented in the Group “1.1.3. Conical surfaces”) The Morse cone (synonym is conical surface of revolution) Oblique elliptical conical surface (it is presented in the Group “1.1.3. Conical surfaces”) Parabolic conical surface (it is presented in the Group “1.1.3. Conical surfaces”) Hyperbolic conical surface (it is presented in the Group “1.1.3. Conical surfaces”) Elliptical cylinder (it is presented in the Group “1.1.2. Cylindrical surfaces”) Hyperbolic cylinder (it is presented in the Group “1.1.2. Cylindrical surfaces”) Parabolic cylinder (it is presented in the Group “1.1.2. Cylindrical surfaces”) Oblique circular cylinder (it is presented in the Group “1.1.2. Cylindrical surfaces”) Oblique elliptical cylinder (it is presented in the Group “1.1.2. Cylindrical surfaces”) Oblique hyperbolic cylinder (it is presented in the Group “1.1.2. Cylindrical surfaces”) Oblique parabolic cylinder (it is presented in the Group “1.1.2. Cylindrical surfaces”) ■ Quadrics

36. Algebraic Surfaces of the High Orders

■ Algebraic surfaces of the high orders presented in the encyclopedia

36.1. Algebraic Surfaces of the 3th Order

■ “Handkerchief surface” ■ Cubic surface x3 +xy+z=0 ■ Diagonal cubic surface of Fermat ■ Cubic cone ■ The cubic surface with 24 straight lines

36.2. Algebraic Quartic Surface

■ Euler surface of the forth order ■ Surface of circles of Feuerbach ■ Nordstrand’s weird surface ■ Menn’s surface ■ Środa’s parabolic surface ■ Overlimited parabolic velaroid ■ “Chair” ■ Goursat’s surface ■ Tooth surface ■ Tanglecube ■ Kummer surface ■ Quartic surface with parabola, ellipse, parabola in three principal coordinate sections ■ Quartic surface with the 4th order curve, ellipse, the 4th order curve in three principal coordinate sections ■ Pillow Shape ■ Additional information on surfaces of the 4th order Classification of All Surfaces Presented in the Encyclopedia 717

36.2.1. The 4th Order Surfaces With Multiply Lines ■ Quartic surface with a triple straight line ■ Quartic surface with a double conic and with a double straight line ■ Quartic surface with two double straight lines

Quartics with a double conic

Dupin cyclides (they are presented in the Group “17.1.2. Cyclic surfaces”) ■ Cyclides with a triple point

Quartics with three double straight lines

■ Ruled surfaces of the 4th order with three double lines ■ Steiner surfaces of the ﬁrst and second types

Quartics with a double straight line

■ Quartic surface with a double straight line and a triple point

36.3. Algebraic Quintic Surfaces

■ Peninsula surface ■ Parabolic surface of conoidal type

36.4. Algebraic Surfaces of the Sixth Order

■ Sine surface ■ Heart surface ■ Hunt‘s surface ■ The 6th order surface with the pseudo Agnesi curl and two parabolas lying in parallel planes ■ The 6th order surface with parabola, the 4th order curve, parabola lying in three principal coordinate sections ■ Surface of revolution of an Agnesi curl (the 2nd variant) ■ Algebraic surface of the 6th order with two nets of translation ■ The 6th order surface with Agnesi curl, ellipse, Agnesi curl lying in three principal coordinate sections

36.5. Algebraic Surfaces of the 7th Order

■ “Ski hill” ■ The 7th order surface with parabola, Agnesi curl, ellipse lying in three principal coordinate sections ■ The 7th order surface with parabola, ellipse, Cartesian folium lying in three principal coordinate sections ■ The 7th order surface with parabola, the 4th order curve, parabola lying in three principal coordinate sections ■ The 7th order surface with Agnesi curl, Lame’s curve of the third order, straight lines lying in three principal coordinate sections

36.6. Algebraic Surfaces of the 8th Order

■ The 8th order surface with Agnesi curl, ellipse, ellipse lying in three principal coordinate sections ■ The 8th order surface with Lame’s curve of the 4th order, Lame’s curve of the 4th order, ellipse lying in three principal coordinate sections ■ The 8th order surface with parabola, the 4th order curve, parabola lying in three principal coordinate sections 718 Classification of All Surfaces Presented in the Encyclopedia

36.7. Algebraic Surfaces of the 9th Order

Enneper’s surface (it is preented in the Class “19. Minimal surfaces”)

36.8. Algebraic Surfaces of the 10th Order

Aerodynamic surfaces given by a continuous framework of elliptical ribs (it is presented in the Subclass “6.1. Aerodynamic surfaces given by algebraic plane curves”)

36.9. Algebraic Surfaces of the 12th Order

■ Surface of the 12th order with parabola, the 8th order curve, parabola lying in three principal coordinate sections

36.10. Algebraic Surfaces of the 16th Order

■ Algebraic surface with a continuous net of the pseudo Agnesi curl passing through a parabola and two straights The literature on geometry of algebraic surfaces

37. Polyhedrons and Quasi-Polyhedrons

■ The types of polyhedrons ■ Astroidal ellipsoid ■ Hyperbolic octahedron ■ The models of quasi-polyhedrons presented in sites of Internet

38. Equidistances of Double Systems

■ Equidistance of the system “straight–cylinder” ■ Equidistance of the system “straight–sphere” ■ Equidistance of the system “point–cylinder” ■ Equidistance of the system “straight–torus”

The Surfaces Not Included in the Classification

Pontryagin’s surfaces Kuen surfaces Bress hyperboloid Euler special cones Surfaces with the constant equi-afﬁne invariants Steiner ellipsoid Ortoid ruled surfaces Algebraic congruence of the 4th order of the 2nd class Fractal surfaces Russian, English, French, and German Dictionary of Surfaces and Curves

See Table 1

Table 1 Russian English French German Aбeлeвa минимaльнaя Abelian minimal surfaces Abelsche Minimalflächen пoвepxнocть Aлгeбpaичecкaягeoмeтpия Algebraic geometry géométrie f algébrique f algebraische Geometrie f aлгeбpaичecкaяпoвepxнocть Algebraic surface surface f algébrique algebraische Fläche f aнaлитичecкaяпoвepxнocть Analytical surface surface f analitique Analytische Fläche f acимптoтa Asymptote of the curve asymptote f Asymptote f acимптoтичecкaя линия Asymptotic line ligne f asymptotique Schmiegtangentkurve f acтpoидa Astroid astroïde f Astroide f aффиннo-минимaльнaя Affine minimal surface surface minimale affine Affinminimalfläche f пoвepxнocть aффиннo-минимaльныe Affine minimal surfaces of surface minimale affine de Die Affinminimalflächen von G. пoвepxнocти Toмceнa Thomsen Thomsen Thomsen Бaшмaчнaяпoвepxнocть Shoe surface surface souliève Schuhfläche f бинopмaль Binormal binormale f Binormale f Бoгeмcкий кyпoл Bohemian dome Dôme Bohemian бyтылкa Клeйнa Klein bottle la bouteille de Klein бyтылкa Уитни Whitney bottle la bouteille de Whitney Beктop Vector vecteur m Vektor m вeлapoидaльнaяпoвepxнocть Velaroidal surface surface vélaroidale Bepзиepa (лoкoн Aньeзи) Agnesi curl boucle Agnesi вepшинa кoнyca Vertex of a cone sommet m d’un cône Spitze f, Scheitel m винт Apxимeдa Archimedes’ screw vis d’ Archimède Archimedische Schnecke винтoвaя линия Helix, coil helice f Schraubenlinie f винтoвaяпoвepxнocть Helical surface surface f hélicoïdale Schraubfläche винтoвaятpyбчaтaя Helical tubular surface surface f hélicoïdale tubulaire пoвepxнocть винтoвoe движeниe Screw motion mouvement m hélicoïdal schraubenförmige Bewegung винтoвoйкoнoид Helical conoid conoïde hélicoïdal (continued)

S.N. Krivoshapko and V.N. Ivanov, Encyclopedia of Analytical Surfaces, 719 DOI 10.1007/978-3-319-11773-7, © Springer International Publishing Switzerland 2015 720 Russian, English, French, and German Dictionary of Surfaces and Curves

Table 1 (continued) Russian English French German винтoвoй Helical cylindroid cylindroïde hélicoïdal цилиндpoид винт Штeйнбaxa Steinbach screw vis de Steinbách вoлнoвaяпoвepxнocть Fresnel’s elasticity surface surface ondulatoire de Fresnel Фpeшнeлa втopaя ocнoвнaя The second fundamental form of a seconde forme f quadratique zweite Fundamentalform f einer квaдpaтичнaяфopмa surface fondamentale d’une surface Fläche пoвepxнocти вытянyтый эллипcoид Prolate spheroid ellipsoïde m de revolution allongé langgestrecktes вpaщeния Rotationsellipsoid n Гayccoвa кpивизнa The Gaussian curvature of surface courbure f de Gauss Gauβsche Krümmung f пoвepxнocти гeкcaэдp Hexahedron hexaèdre m Hexaeder n гeликoид Дини Dini’s surface hélicoïde Dini Dinische Helicoid n гeoгpaфичecкaя cиcтeмa Geographic system of curvilinear système géographique des кpивoлинeйныx кoopдинaт coordinates coordonnées curvilignes гeoдeзичecкaякpивизнa Geodesic curvature courbure f géodesique geodätische Krümmung f гeoдeзичecкaя линия Geodesic line ligne géodésique geodätische Linie f гипap Hypar hypare гипepбoлa Hyperbola hyperbole f Hyperbel m гипepбoличecкиe Hyperbolic coordinates coordonnèes fplhyperboliques hyperbolische Koordinaten fpl кoopдинaты гипepбoличecкий гeликoид Hyperbolic helicoid hélicoïde hyperbolique hyperbolisches Helikoid гипepбoличecкий Hyperbolic paraboloid paraboloïde m hyperbolique hyperbolisches Paraboloid n пapaбoлoид (гипap) гипepбoличecкий тopc Hyperbolic developable surface surface éveloppable hyperbolique гипepбoличecкий цилиндp Hyperbolic cylinder cylindre m hyperbolique hyperbolischer Zylinder гипoциклoидa Hypocycloid hypocycloïde f Hypozykloide f гипoциклoидный цилиндp Hypocycloidal cylinder hypozykloid Zylinder глaвнaянopмaль Principal normal normale principale Hauptnormale f глaвнaякpивизнa Principal curvature courbure f principale Hauptkrümmung f глaвныe кpивизны Principal curvatures of surface surface d’incurvation principale Hauptkrümmung f пoвepxнocти глaдкaякpивaя Smooth curve courbe f lisse glatte Kurve f глoбoид Globoid globoïde Globoid гopлoвaя oкpyжнocть Waist circle Kehlekreis f гopлoвoй эллипc Waist ellipse Kehleellipse f гoфp Crimp Двaжды линeйчaтaя Doubly ruled Surface surface double règlèe пoвepxнocть двyпoлocтный гипepбoлoид Hyperboloid of two sheets hyperboloïde m deux nappes zweischaliges Hyperboloid n двyпoлocтный гипepбoлoид Two-sheeted hyperboloid of hyperboloïde à deux nappes de zweischaliges Drehhyperboloid n вpaщeния revolution rotation дeкapтoвы кoopдинaты Cartesian coordinates coordonnées fplcartésiennes kartesische Koordinaten fpl диффepeнциaльнaя Differential geometry géométrie f différentielle Differentialgeometrie f гeoмeтpия длинa дyги Arc length longueur f d’un arc Bogenlänge f дoдeкaэдp Dodecahedron didécaèdre m Dodekaeder n (continued) Russian, English, French, and German Dictionary of Surfaces and Curves 721

Table 1 (continued) Russian English French German Зaкpытый тop Spindle torus; horn torus tore fermé geschlossener Torus m зaмкнyтaяпpocтpaнcтвeннaя Closed space curve courbe fermée gauche räumliche abgeschlossene Kurve кpивaя зoнтик Кapтaнa Cartan umbrella parapluie de Cartan зoнтик Уитни Whitney umbrella parapluie de Whitney зyбнaяпoвepxнocть Tooth surface Изгибaниe Bending déformation f Verbiegung f изoмeтpия Isometry, isometric projection isométrie f isometrische Abbildung икocaэдp Icosahedron icosaèdre m Ikosaeder n Кaнoничecкoe ypaвнeниe The support function of the equation canonique de surface kanonische Gleichung einer пoвepxнocти surface Oberfäche кaнaлoвaяпoвepxнocть Canal surface of Joachimsthal surface canale de Joachimsthal Иoaxимcтaля кaплeвиднaяпoвepxнocть Drop shaped surface surface de la goute кacaтeльнaяплocкocть Tangent plane plan m tangent Tangentialebene f кaтeнoид Catenoid caténoïde f Katenoide f квaдpaтичнaяпoвepxнocть Quadric surface surface quadratique quadratische Fläche f клacc Class classe f Klasse f кoнгpyэнтнocть Congruence congruence f Kongruenz f кoникa Conic conique кoничecкaя винтoвaя линия Conical helix, conic spiral spirale f conique konische Schraubenlinie кoничecкaякpoмкa Уoллиca Wallis’s conical edge bord conique de Wallis кoничecкaяпoвepxнocть Conic surface of revolution surface conique de revolution Drehkegelfläche f вpaщeния кoничecкaя cпиpaль Conical spiral spirale conique кoничecкий гeликoид Conic helicoid of constant slope hélicoïde conique de pente oдинaкoвoгocкaтa constante кoнoид Conoid conoïde m Konoid n, Konoidfläche f кoнoид Плюккepa Plücker conoid conoïde Plücker Plückersches Konoid кoнoид Циндлepa Zindler’s conoid conoïde de Zindler Zindlersches Konoid кoнyc Cone cône m Kegel m кocoйгeликoид Oblique helicoid hélicoïde m oblique кocoй гипap Oblique hypar hypare oblique кoэффициeнты пepвoй Coefficients of the first coefficients de première forme квaдpaтичнoйфopмы fundamental form of surface quadratique de surface пoвepxнocти кoэффициeнты втopoй Coefficients of the second coefficients de deuxième forme квaдpaтичнoйфopмы fundamental form of surface quadratique de surface пoвepxнocти кoэффициeнты Лaмe в Lame’s coefficients in the surface coefficient de Laméàla théorie des тeopии пoвepxнocтeй theory surfaces кpaтныe линии Multiple lines lignes multiples кpecтooбpaзный жeлoб Crossed trough canal croix кpивaявтopoгo пopядкa Quadratic curve, conic courbe de deuxiéme orde, conique f Kurve zweiter Ordnung, (кoникa) Kegelschnitt кpивaяЛaмe Lame’s curve courbe de Lamé Lamesche Kurve f кpивaя 3-гo пopядкa Cubic (curve) courbe d’ordre trois, cubique f Kurve dritter Ordnung, Kubik f (continued) 722 Russian, English, French, and German Dictionary of Surfaces and Curves

Table 1 (continued) Russian English French German кpивизнa кpивoй Curvature of the curve courbure f de la ligne Krümmung der Kurve кpивoлинeйныe кoopдинaты Curvilinear coordinates on the coordonnées fplcurvilignes de la krummlinige Koordinaten fpl нa пoвepxнocти surface surface кpyгoвaяпoвepxнocть Circular translation surface surface de translation curculaire пepeнoca кpyгoвoйкoнyc Circular cone cône m à base circulaire Kreiskegel m кpyгoвoйтop Circular torus tore circulaire Kreistorus m кpyгoвoй цилиндp Circular cylinder cylindre m de revolution Kreiszylider m, Drehzylinder m кyб Cube cube m Würfel m кyбичecкaякpивaя (кyбикa) Cubic curve (cubic) cubique f Kubik f кpyчeниe кpивoй Torsion of the curve courbe de torsion Schmiegung f einer Raumkurve кyбичecкий тopc Cubic developable surface surface développable cubique кyбoид Cuboid cuboïde m Лeмниcкaтa Lemniscate lemniscate f Lemniskate f Лимoн Lemon surface Die Lemon Surface линeйчaтaя винтoвaя Ruled helical surface Regelschraubfläche f пoвepxнocть линeйчaтaяпoвepxнocть Ruled surface, scroll surface, surface f réglée Regelfläche f scroll линия кpивизны Line of curvature ligne f de courbure Krümmungslinie f линия oткoca Sloping line, line of slope линия цeнтpoв oкpyжнocтeй The line of centers of circles, ligne des centres des cercles die Linie der Kreismittelpunkt center-to-center line ЛиcтДeкapтa Cartesian folium (leaf) лиcт Meбиyca Möbius strip ruban m de Moebius Möbiussches Band n лoгapифмичecкaя cпиpaль Logarithmic spiral spirale f logarithmique logarithmische Spirale f лoкoн Aньeзи Agnesi curl boucle Agnesi лoкcoдpoмa Loxodrome loxodromie f Loxodrome f Mepидиaн Meridian méridien m Meridian m Mepкaтopoвa пpoeкция The Mercator parameterization projection de Mercator минимaльнaяпoвepxнocть Minimal surface surface f minimale Minimalfläche f минимaльнaяпoвepxнocть Bour’s minimal surface surface minimale de Bour Bohrsche Minimalfläche Бopa минимaльнaяпoвepxнocть Costa minimal surface surface minimale de Costa Кocты Mинимaльнaяпoвepxнocть Thomsen’s minimal surface surface minimale de Thomsen Thomsensche Minimalfläche f; Toмceнa Minimalfläche f von G. Thomsen минимaльныe пoвepxнocти Minimal surfaces with free co cвoбoдными гpaницaми boundaries мнoгoгpaнник Polyhedron polyèdre m Polyeder n, Vielflach n мopcкaя paкyшкa seashell coquille f Haклoннaякoничecкaя Oblique conic surface surface conique inclinée пoвepxнocть нaклoнный цилиндp Oblique cylinder cylindre incliné нaпpaвляющaякpивaя Directrix curve courbe guide Leitkurve f нaпpaвляющиe кocинycы Directional cosines нaпpaвляющий кoнyc Director cone (continued) Russian, English, French, and German Dictionary of Surfaces and Curves 723

Table 1 (continued) Russian English French German нaчepтaтeльнaягeoмeтpия Descriptive geometry géométrie f descriptive f darstellende Geometrie f нeпpepывный кapкac A continuous net of rectilinear carcasse continue des génératrices пpямoлинeйныxoбpaзyющиx generatrixes of developable rectilignes de surface développable тopca surface нopмaльнaяплocкocть Normal plane of space curve plan normal de courbe spatiale Normalebene f der raumlichen пpocтpaнcтвeннoйкpивoй Kurve нopмaльнaя цикличecкaя Normal cyclic surface surface normale cylindrique пoвepxнocть Oбeзьяньeceдлo Monkey saddle celle de singe Affenstattel oбoбщeнный Generalized généralisé verallgemeinert oбpaзyющaя oкpyжнocть Generating circle cercle génératrice erzeugende des Kreislinie oбычнaя циклoидa Ordinary cycloid cycloïde ordinnaire oднoпapaмeтpичecкoe Single-parametric System of famille des plans à un parametre ceмeйcтвo плocкocтeй planes oднoпoлocтный гипepбoлoид Hyperboloid of one sheet hyperboloïde m à une nappe einschaliges Hyperboloid n oднoпoлocтный гипepбoлoид One-sheet hyperboloid of hyperboloïde de revolution à une einschaliges Drehhyperboloid вpaщeния revolution nappe oкpyжнocть Circle cercle m Kreislinie f oкpyжнocти Bиллapco cercles de Villarceau oктaэдp Octahedron octaèdre m Oktaeder n oмбиличecкaяпoвepxнocть Umbilical surface surface ombilique Nabelﬂäche f ocнoвaниe цилиндpa Base of cylinder cylindre de base ocoбaятoчкa Singular point point m singulier singulärer Punkt m ocь Axis axe m Achse f ocь aбcциcc x-axis, abscissa axis axe d’abscisses Abszissenache f ocь aппликaт z-axis, applicate axis axe de cotes, axe des Z Applikatenachse f ocьвpaщeния Rotation axis axe de rotation ocь opдинaт y-axis, ordinate axis axe d’ordonnées Ordinatenachse f ocь cиммeтpии Axis of symmetry axe m de symétrie Symmetrieachse f oткpытый тop Ring torus tore ouvert Пapaбoлa Parabola parabole f Parabel f пapaбoлaHeйля Neil’s parabola parabole f de Neil Neilsche Parabel f пapaбoличecкий вeлapoид Parabolic velaroidal surface surface vélaroïdale parabolique пapaбoличecкий кoнoид Parabolic conoid conoïde parabolique пapaбoлoид вpaщeния Paraboloid of revolution paraboloïde de révolution Rotationsparaboloid n пapaбoличecкaятoчкa Parabolic point point parabolique parabolischer Punkt пapaбoличecкий тopc Parabolic developable surface surface développable parabolique пapaбoличecкий цилиндp Parabolic cylinder cylindre m parabolique parabolischer Zylinder m пapaллeль Parallel parallele Parallele f пapaмeтpичecкиe ypaвнeния Parametric equations of surface equations paramétrique de surface пoвepxнocти пeнкa Skin plau f пepвaя ocнoвнaя The ﬁrst fundamental form of a première forme quadratique erste Fundamentalform f einer квaдpaтичнaяфopмa surface fondamentale d’ une surface Fläche пoвepxнocти пepeкpyчeннaялeнтa Twisting ribbon ruban forcéüberdrehendes Band (continued) 724 Russian, English, French, and German Dictionary of Surfaces and Curves

Table 1 (continued) Russian English French German Пиpeнeйcкaяпoвepxнocть Peninsula surface surface Peninsula плocкocть Plane plan m Ebene f плoщaдь пoвepxнocти The surface area l’aire de la surface плyжный oтвaл Mouldboard versoir de charrue Haldenpflug m пoвepxнocти Бeзьe Surfaces of Bézier surfaces de Bézier Bézier-Flächen пoвepxнocти Блютeля Blutel’s surfaces surfaces de Blutel Die Blutelschen Kegelschnittflächen пoвepxнocти Бoннe Bonnet surfaces surfaces d’Ossian Bonnet пoвepxнocти Beйнгapтeнa Weingarten surfaces surfaces de Weingarten W-Flächen пoвepxнocти Bepoнeзe Veronese surfaces surfaces de Veronese пoвepxнocти Кaтaлaнa Catalan’s surfaces surfaces de Catalan Katalanischen Flächen Пoвepxнocти Кyнca Coons surfaces пoвepxнocти Пeтepcoнa Peterson surfaces surfaces de Peterson Petersonischen Flächen пoвepxнocти Xaкeнa Haken surfaces surfaces de Haken Hackenischen Flächen пoвepxнocти Цицeйки Tzitzéica’s surfaces surfaces de Tzitzéica пoвepxнocти Эдлингepa Edlinger’s surfaces surfaces d’Edlinger der Edlinger-Flächen пoвepxнocть Бoя Boy surface surface f de Boy Boyische Fläche пoвepxнocть вpaщeния Surface of revolution surface f de révolution Rotationsfläche f пoвepxнocть вpaщeния Surface of revolution of astroid surface de revolution des astroïdes Rotationsfläche der Astroide acтpoиды пoвepxнocть вpaщeния Funnel Der Funnel «Bopoнкa» пoвepxнocть вpaщeния The surface of revolution of surface semi-cubique des paraboles Rotationsfläche der пoлyкyбичecкoйпapaбoлы semicubical parabola de revolution semikubische Parabel пoвepxнocть вpaщeния Surface of revolution of cycloid surface de revolution cycloïde Rotationsfläche der Zykloide циклoиды пoвepxнocть втopoгo пopядкa Quadratic surface surface quadratique zweiten Grades Fläche f пoвepxнocть Гypca Goursat’s surface surface de Goursat Gurssche Fläche пoвepxнocть Дapбaзи Darbazi surface surface de Darbazi Darbazische Fläche пoвepxнocть двaжды Surface of a double oblique surface d’un coin double-oblique кocoгoклинa wedge пoвepxнocть двaжды кocoгo Surface of a double oblique surface d’un cylindre double- цилиндpa cylinder oblique пoвepxнocть Зeйфepтa Seifert surface surface de Seifert пoвepxнocть Кyммepa Kummer surface surface de Kummer пoвepxнocть Кyeнa Kuen surface surface de Kuen пoвepxнocть Кэли Cayley surface surface de Cayley Cayleyfläche пoвepxнocть кocoгo клинa Surface of an oblique wedge surface d’un coin oblique Пoвepxнocть «Кpecлo» “Chair” surface “Chaise” Пoвepxнocть кpyгoв Surface of circles of Feuerbach Фeйepбaxa пoвepxнocть Meбиyca Möbius surface surface de Moebius пoвepxнocть Mopэнa Morin’s surface surface de Morin пoвepxнocть кocoгo Surface of an oblique cylinder surface d’un cylindre oblique цилиндpa пoвepxнocть Mэннa Menn’s surface surface de Menn (continued) Russian, English, French, and German Dictionary of Surfaces and Curves 725

Table 1 (continued) Russian English French German пoвepxнocть oдинaкoвoгo Surface of constant slope surface de pente constante cкaтa пoвepxнocть «Пaдaющaя “Kiss surface” кaпля» пoвepxнocть «Плaтoк» “Handkerchief surface” пoвepxнocть пpямoгo Translation surface surface de translation пepeнoca пoвepxнocть пятoгo пopядкa Quintic surface пoвepxнocть Peмбca Rembs’ surface surface de Rembs пoвepxнocть «Poгизoбилия» “Cornucopia”“Corne d’Abondance” Пoвepxнocть cвятoгo Ильи “The Saint Elias surface” Пoвepxнocть «Cepдцe» Heart surface surface «Coeur» пoвepxнocть Cивepтa Sievert’s surface surface de Sievert пoвepxнocть Xaнтa Hunt’s surface surface de Hunt пoвepxнocть чeтвepтoгo Quartic surface; the 4th order surface quatrique пopядкa surface пoвepxнocть Шepкa Scherk’s minimal surface surface minimale de Scherk Die Scherksche Minimalfläche пoвepxнocть Штeйнepa Steiner surface surface de Steiner Steiner Fläche f пoвepxнocть Эннeпepa Enneper’s surface surface d’Enneper Die Ennepersche Minimalfläche пoлинoмиaльнaяфyнкция Polynomial function fonction polynomiale Polynom m пoлнaя минимaльнaя Complete minimal surface пoвepxнocть пoлyкyбичecкaяпapaбoлa Semicubical parabola (cuspidal parabole f semi-cubique semikubische Parabel f cubic) пoлyocи Semi-axes demi-axes пoляpнaяпoвepxнocть Polar surface surface polaire Polfläche f пoляpныe кoopдинaты Polar coordinates coordonèes fplpolaires polare Koordinaten fpl пpaвильный гипepбoлoид Regular hyperboloid hyperboloïde régulier пpeдмeтнaяплocкocть Object plane plan object Grundebene пpизмa Prism prisme m Prisma n пpизмa yceчeннaя Truncated prism tronc m de prisme, prisme Prismenstumpf m m tronqué пpoлeт oбoлoчки Span travée Feld пpямoйгeликoид Right helicoid helicoïde droit пpямoй гипap Right hypar hypare droit пpямoй цилиндpoид Right cylindroid cylindroïde droit пpямoлинeйнaя oбpaзyющaя Rectilinear generatrix generatrice rectiligne пpямoyгoльныe кoopдинaты Orthogonal coordinates coordonnées fplrectangulaires rechtwinklige Koordinaten fpl пceвдocфepa Pseudosphere, tractroid, pseudo-sphère f Pseudosphäre f tractrisoid Paдиyc Radius rayon m Radius m paдиyc-вeктop Radius-vector rayon m vecteur Radiusvektor m paзвepтывaющaяcя Developable surface surface f dévelopable abwickelbare Fläche f пoвepxнocть paзвepтывaющийcягeликoид Open helicoid hélicoïde ouvert abwickelbare Helykoid paзвepтывaющийcя Developable conic helicoid hélicoïde développable conique abwickelbare Kegelhelikoid кoничecкий гeликoид (continued) 726 Russian, English, French, and German Dictionary of Surfaces and Curves

Table 1 (continued) Russian English French German paзвepткa Development developpement m entwickelte Fläche f peбpo вoзвpaтa Cuspidal edge l’arête de rebroussement Gratlinie f, Striktionslinie peгyляpнaяпoвepxнocть Regular surface surface reguliére peзнaя линeйчaтaя Monge ruled surface surface regleé de Monge пoвepxнocть Moнжa peзныe пoвepxнocти Moнжa Monge surface of double surfaçes moulures Gesimsflächen двoйнoйкpивизны curvature pимcкaяпoвepxнocть Roman surface surface romaine pyлeттa Roulette roulette f Rollkurve f Ceдлo Пeaнo Peano saddle celle de Peano ceть пepeнoca Translation net cжaтый эллипcoид вpaщeния Oblate spheroid ellipsoïde m de revolution aplati zusammengedrücktes Rotationsellip-soid n cинycoвaяпoвepxнocть Sine surface surface sunicoidale cинycoидa Sine curve, sinusoid sinusoïde f Sinuskurve f cкpeщeнный кoлпaк Steiner cross cap Штeйнepa coпpикacaющaяcяплocкocть Osculating plane plan m osculateur Schmiegungsebene f coпpяжeнныe Conjugate curvilinear coordinates coordonnées curvilignes кpивoлинeйныe кoopдинaты conjuguées coтoвaяпoвepxнocть Honeycomb surface cпиpaль Apxимeдa Spiral of Archimedes spirale f d’Archimède Archimedische Spirale f cпиpaльнaяпoвepxнocть Spiral surface Spiralfläche cпиpaль Фepмa Spiral of Fermat Fermat Spirale f cплющeнный эллипcoид Oblate ellipsoid of revolution ellipsoïde m de révolution aplati zusammengedrücktes вpaщeния Rotationsellipsoid n cпpямляющaяплocкocть Rectifying plane plan m rectifiant rektifizierende Ebene f cpeдиннaяпoвepxнocть Middle surface of a shell surface moyenne d’une enveloppe Schalenmittelfläche f oбoлoчки cpeдняя кpивизнa The mean curvature of surface courbure f moyenne mittlere Krümmung f der Fläche пoвepxнocти cтpeлa пoдъeмa Rise fléche, montéed’enveloppe Pleilhöhe f cтpикциoннaя линия Striction curve cфepa Sphere sphère f Sphäre f cфepичecкиe кoopдинaты Spherical coordinates coordonnées fplspherique Kugelkoordinaten fpl cфepoид Spheroid sphéroïde m Sphäroid, Rotationsellipsoid Teтpaэдp Tetrahedron tétraèdre m Tetraeder n тop Torus tore m Torus m тopoид Toroid toroïde m Toroid n тopc Torse surface développable Torse тopc 4-гo пopядкa The forth order torse surface surface developpable de 4-ème ordre тoчкa вoзвpaтa пepвoгopoдa The first type cuspidal point point m de rebroussement Rückkehrpunkt m, erster Art тoчкa нa бecкoнeчнocти A point at infinity point m à l’infini Fernpunkt m тoчкa пинчa Pinch point point de pinch тpaктpиca Tractrix tractrice f Traktrix f (continued) Russian, English, French, and German Dictionary of Surfaces and Curves 727

Table 1 (continued) Russian English French German тpaктpoид Tractroid tractroïde m тpaнcцeндeнтнaя Transcendental тpexocный эллипcoид Triaxial ellipsoid ellipsoïde triaxiale тpинoид Trinoid trinoïde m тpoxoидa Trochoid trocoïde f Trochoide f тpyбчaтaяпoвepxнocть Tubular surface, tube surface tubulaire Röhrenfläche f Угoл Angle angle m Winkel m yлиткa Пacкaля Limaçon of Pascal limaçon m de Pascal Pascalsche Schnecke f yндyлoид Unduloid, onduloid Unduloid ypaвнeниe внeявнoйфopмe Implicit equation èquation f implicite implizite Gleichung f ypaвнeниe в явнoйфopмe Explicit equation èquation f explicite explizite Gleichung f yceчeнный кoнyc Truncated cone; conical frustum tronc m de cône Kegelstupf m ycлoвиeeдинcтвeннocти Condition of uniqueness of the condition d’unicité de surface тopcoвoйпoвepxнocти developable surface développable ycлoвиe тopcoвocти Necessary condition for the condition nécessaire pour la surface developable surface développable Фигypa Figure, configuration figure f Figur f фopмyлa Formula formule f Formel f фyнкция Function fonction f Funktion f Xopдa Chord corde f Sehne f Цeпнaя линия Catenary caténaire f Kettenlinie f циклидacтpoйнoйтoчкoй Cyclide with a triple point cyclide avec le triple poïnt циклиды Дюпeнa Dupin’s cyclides cyclide de Dupin цикличecкaяпoвepxнocть Cyclic surface surface cyclique циклoидa Cycloid cycloide f Zykloide f цилиндpичecкaя винтoвaя Circular helix, Cylindrical helix hélice f cylindrique gemeine Schraubenlinie f линия цилиндpичecкaя винтoвaя Cylindrical helical strip barre cylindrique hélicale пoлoca цилиндpичecкaя линeйчaтaя Cylindrical ruled rotational limaçon cylindique réglé rotational poтaтивнaя yлиткa limaçon цилиндp Cylinder cylindre m Zylinder m цилиндpичecкaяпoвepxнocть Cylindrical surface surface f cylindrique Zylinderfläche f цилиндpичecкaяпoвepxнocть Cylindrical surface of revolution surface de revolution cylindrique вpaщeния цилиндpичecкиe кoopдинaты Cylindrical coordinates coordonnèes fplcylindriques zylindrische Koordinaten fpl цилиндpoид Cylindroid cylinroïde n Zylindroid n Чeбышeвcкaя ceть нa Chebyshev’s net on the surface пoвepxнocти Шaг винтoвoй линии The lead of a helix Шaг peзьбы The pitch of a screw шap Ball, solid sphere boule f Kugel m шyтoвcкoйкoлпaк Dunce hat bonnet de bouffon Эвoльвeнтaoкpyжнocти Evolvent of the circle évolvente f circulaire Evolvente f ………. эвoльвeнтный гeликoид Evolvent helicoid hélicoïde d’évolvente эллипc Ellipse ellipse f Ellipse f (continued) 728 Russian, English, French, and German Dictionary of Surfaces and Curves

Table 1 (continued) Russian English French German эллипcoид вpaщeния Spheroid (ellipsoid of revolution) ellipsoïde m de revolution Rotationsellipsoid n, Drehellipsoid эллиптичecкий вeлapoид Elliptic velaroidal surface surface elliptique vélaroidale эллиптичecкий гeликoид Elliptic helicoid hélicoïde elliptique эллиптичecкий кoнyc Elliptic cone cône elliptique эллиптичecкий пapaбoлoид Elliptic paraboloid paraboloïde m elliptique elliptisches Paraboloid n эллиптичecкий цилиндp Elliptic cylinder cylindre m elliptique elliptischer Zylinder m эпитpoxoидa épitrohoïde эпитpoxoидaльнaя surface épitrohoïde пoвepxнocть эпициклoидa Epicycloid épicycloïde f Epizykloide f эпициклoидный цилиндp Epicycloidal cylinder epizykloid Zylinder Яйцo Дyнca Duns egg oeuf de Duns PS The encyclopedia contains a large number of additional literature and references written in Russian language. That is why, the ﬁrst column in the Dictionary is given for Russian terms. It must help to read the presented Russian original sources Name Index of the Basic Text

The names and initials of the scientists used in the basic text of the encyclopedia are given in the Name Index of the Basic Text. Sometimes these names go down in the names of surfaces, constants, or coefﬁcients. After the names of scientists at the same line, the pages of the encyclopedia are pointed out where these names were used.

A Brankov, G., 501 Abresch, U., 483 Brauner, G., 691 Afwat, M., 472 Brauner, H., 545 Agnesi, M., xxvi, 66, 175, 219, 504, 506 van-Brunt, B., 471 Aleksandrov, A.D., 483 Burkhard, Heinrich, 627 Alimov, R.U., 18 Ampere, André-Marie, 319, 474 Annaberdyev, E., 148 C Archimedes, xxiv, xxvi, 6, 51 Calabi, Eugenio, 450 Avdon’ev, E.A., 667, 671 do Carmo M., 483 Cartan, E., 1, 544 Cassini, Giovanni Domenico, 217 B Catalan, Eugène Charles, 1, 77, 92, 213, 427, 435 Ball, Robert Stawell, 94 Cauchy, Augustin Louis, 146, 551 Barbosa, J.L., 483 Cayley, A., 165, 627, 632, 649 Barile, Margherita, 452 Chebotaryov, N.G., 160 Barr, Alan H., 332 Chebyshev (Tchebyshev), P.L., 159, 479 Barthel, W., 451 Chen, C.C., 449 Beltrami, E., 108, 479 Chen, W., 544 Bergmann, Horst, 691 Cherdyntzev, N.V., 151 Bernstein, S.N., 327, 548 Christoffel, Elwin Bruno, xxvii Bernoulli, Jacob, 49 Christopher, Wren, 621 Bernoulli, Johann, 46, 50, 217, 218 Clairaut, Alexis-Claude, 147 Bézier, P., 327, 328, 548 Clebsh, A., 627 Bhattacharya, B., 4 Codazzi, D., xxvii, 150 Bianshi, L., 325 Coffman, A., 421, 654 Bîlă, N., 319–321 Coon, Dan, 547 Björling, Emanuel Gabriel, 440 Coons, Steven Anson, 547–550 Blaschke, W., 451 Costa, A., 437 Blum, Richard, 543 Costa, Celso da, 437 Blutel, 315 Cotes, Roger, 50 Bodunov, A.K., 148 Courante, F., 445 Bodunov, N.A., 148 Bonnet, O., xvii, 427, 543, 544 Bottema, O., 691 D Bour, E., 226, 436 Darboux, xvii, 249, 625 Boy, W., 419, 422, 424 Dedonder, W., 419 Bracken, Paul, 480 Degen, W., 315 Brakke, K., 455 Dehtyar A.S., 148 Brandt, G.V., 142 Delaunay, C., 155

S.N. Krivoshapko and V.N. Ivanov, Encyclopedia of Analytical Surfaces, 729 DOI 10.1007/978-3-319-11773-7, © Springer International Publishing Switzerland 2015 730 Name Index of the Basic Text

Deriche, Mohamed, 127 Henneberg, L., 439, 440 Descartes, René, 49 Hilbert, D., 417, 479 Dillen, F., 160 Hildebrandt, S., 445 Dini, Ulisse, 239 Hirz, Mario, 229 Djondjorov, P.A., 156 Hoang, Zuy Thang, 658 Đukanović, Gordana, 649 Hoffmann, David, 437, 455 Dupin, Charles, 352, 353, 355, 649 Hoffmann, James, 455 Holgate, Th.F., 649 Hopf, H., 471 E Hunt, Bruce, 661 Edlinger, R., 545, 546 Huygens, Ch., 46 Eisland, J., 663 Eklundh, J.-O., 185 Enneper, Alfred, 339, 432, 451 I Escher, Maurits Cornelis, 358, 417 Ibolya Szilágyi, D.M., 634 Euler, L., xvii, 152, 155, 427, 443, 636, 678 Ignatenko, V.F., 663 Ivanov, V., 226 Ivanov, V.N., 19, 580 F Farshad, M., 148 Fauconnier, M., 82 J von Feuerbach, Karl Wilhelm, 637 Jackway, Paul T., 127 Fepl, Stanimir, 637 Jeener, Patrice, 691 de Fermat, Pierre, 50, 633 Joachimsthal, Ferdinand, 348, 557 Finikov, S.P., 4 Jordan, C., xix Flöry, S., 73 Jorge, L.P.M., 441 Fomina, T.N., 450 Foster, N., 485 Francis, G.K., 692 K Frenet, F., xxii Kalinin, V.S., 150, 151 Freyssinet, Eugène, 82 Kantorovich, A.E., 445 Frezier, Amédée-François, 96 Karcher, H., 443, 449, 454 Kardashevskaya, Yu.G., 9 Kautny, W., 249 G Kenmotsu, K., 156 Gackstatter, Fritz, 449 Kharabayev, Yu.N., 606 Gaudi, Antoni, 87 Kirillov, S.V., 217 Gauss, C.F., xxix, 474 Klein, Felix, 419, 422, 475 Gergonne, G.D., 427, 445 Korolev (Korolyov), E.A., 325, 450 Gielis, J., 537 Kovantzov, N.I., 555 Glogovskiy, V.V., 685 Krames, J.L., 650 Goemans, W., 160 Krasovskiy, F.N., 622 Gordon, 475, 479 Krauter, P., 459 Gorjanc, Sonja, 650, 653 Krivoshapko, S.N., 22, 26 Goursat, E., xvii, 325, 642 Kuen, T., 480, 692 Grant, K., 471 Kühnel, W., 471 Gray, A., 437 Kummer, Ernst Eduard, 645 Grigorenko, Ya.M., 386 Gromov, M.Ya., 418 Groβe-Brauckmann, K., 438 L Gumen, N.S., xvii, 688 Lagrange, G.L., 152, 155 Gurevich, V.I., 150, 151 Lalan, V., 543 van Gutschoven, G., 217 Lamé, G., 2, 15, 332, 536 Laplace, P.S., 148, 480, 551 Legendre, A.M., 427 H Leibniz, G.W., 46 Hahn, L., 85 Li, H., 544 Haken, Wolfgang, 692 Lichtenfels, O., 442 Hashimoto, 480 Lie, S., 159 Haubitz, I., 451 Liebmann, H., 474 Haustova, N.D., 92 Liouville, J., xvii Heaviside, Oliver, 331, 333, 336 Listing, Johann Benedict, 418 Hein, Piet, 333, 536 Lobachevski, N.I., 108 Name Index of the Basic Text 731

Logan, Adam, 649 Roeschel, Ot., 417 López, F.J., 450, 452 Rosenberg, H., 449, 474 Lusta, G.I., 574, 580 Rosendorn, E.R., 563 Roussos, Ioannis M., 543 Runge, C., 445 M Ryzhov, N.N., 18 Mandelbrot, Benoît, 692 Mannheim, 249 Martín, F., 450 S Maryukova, N.E., 475 Sachs, Hans, 545 Meeks, W., 437, 441 Scherk, H.F., 427, 443 Menn, 638 Schaal, Hermann, 451, 452 Meusnier, J.B., 427, 428, 443 Schwartz, H.A., 433, 434, 454 Minding, Ferdinand, 474, 479 Shoen, A.H., 438, 454, 456 Mlodzievskiy, B.K., 325 Shtaerman, Yu.Ya., 182 Möbius, August Ferdinand, 418 Sievert, Herbert, 478 Monge, Gaspard, 3, 185, 186, 189, 190, 193, 195, 196, 319, 427 Simenko, O.V., 651 Morin, Bernard, 692 Skidan, I.A., 196, 498 Müller, E., 650 Smerichko, O.V., 688 Środa, P., 639 Stanislavskaya, L.A., 678 N Steblyanko, V.G., 347 Naeve, A., 185 Stefanova, St., 555 Nankov, G., 226 Steiner, Jakob, 317, 421, 654 Naridashvili, 450 Steller, M., 471 Neile, William, 46, 440 Stolyarchul, V.A., 148 Neovius, E.R., 435 Strubecker, Karl, 459 Newton, Isaac, 217 Sturm, M., 155, 649 Niče, Vilko, xvii Švec, Alois, 317 Nikityuk, V.A., 331 Svetlanova, O.I., 186 Nordstrand, T., 638, 642, 661

T O Takehiro, Iton, 317 Obradović, Marija, 649 Taubin, G., 660 Obukhova, V.S., 10, 73 Thomsen, G., 451, 459, 460 Onischuk, N.M., 186, 691 Togino, Kadzuto, 691 Osserman, R., 428 Togliatti, E.G., 656 Toubiana, Eric, 449 Tzitzéica (Titeica), G., 319–322 P Pascal, Étienne, 347 Palman, D., 691 U Peano, Giusepe, 562 Udrişte, C., 320, 321 Pei, I.M., 54 Peternell, G.M., 343 Peterson, K.M., xxvii, 150, 325, 326, 343, 427, 450 V Pilarski, I., 82 Varignon, Pierre, 50 Plateau, J., 427, 445 Varvaritza, A.G., 35, 214 Platon (Πλάτων), 677 Vasil’eva, O.V., 153 Plucker, J., 89, 91, 95 Veronese, G., 317, 421 Podgorny (Podgorniy), A.L., 73, 658 Villarceau, Antoine-Joseph Yvon, 112 Poisson, S.D., 157, 427 Virich, S.O., 558 Polthier, K., 454 Volkmer, R., 451 Pontryagin, L.S., 691 Volkov, G.F., 176, 177 Popa, Emil M., 319 Vorobkevich, R.I., 10 Popov, A.G., 480 Voronina, A.N., 15 Pottmann, H., 73, 343 Voss, A., xvii Protod’yakonov, S.M., 667

W R Wallis, J., 89, 90, 652 Rachkovskaya, G.S., 606–612 Weierstrass, K., 428, 432, 433, 439, 441, 449, 452, 453 Rembs, E., 327, 476 Weingarten, J., 160, 472, 544 Richmond, Herbert William, 443, 444 Weisstein, Eric W., 437, 441, 443, 449, 478, 627 Riemann, B., 427, 433, 448, 450, 483, 551 Wenninger, M., 679 732 Name Index of the Basic Text

Wente, H.C., 483 Y Werner, A.L., 561 Yadgarov, D.Ya., 580 Whitney, Hassler, 78, 88 Whittemore, J.K., 259 van de Wolstyne, I., 160 Z Wright, F.L., 54 Zaha Hadid, 73 Wunderlich, Walter, 12 Zindler, 91 Name Index of the Authors of Additional Literature

A B Abbena, Elsa, 73 Baba-Hamed, Ch., 258 Abbushi, Nasr Yunes, 342, 520, 523, 524, 559 Babenko, V.I., 4 Abdel-Baky, R.A., 212 Babin, Yu.A., 229 Ablisimov, A.V., 219 Baikoussis, Ch., 263 Abresch, U, 483 Bakel’man, Iya, 561, 565 Adilov, P.O., 674 Balan, V., 320 Afwat, M, 472 Balazs, N.L., 482 Ag’malov, I.R., 457 Baldes, Al., 474 Ahmet’ev, P.M., 424 Ball, Robert Stawel, 95 Ahuja, B.M., 183 Balser, Andreas, 441 Aïssè Gbaguidi, G.L., 34 Banchoff, T., 417 Aleksandrov, A.D., 483, 628 Bandurin, N.G., 144 Aleksandrov, P.V., 257 Bandyopadhyay, J.N., 83 Alimov, R.U., 18, 21, 28 Banichuk, N.V., 149 Al-Bokhary, A.J., 212 Bantlin, A., 343 Altenbach, H., 145 Baranenko, V.A., 378 Amadori, Kristian, 537 Barbagelata, Andrea, 74 Aminov, Yu.A., xxix Barbosa, J Infante, 148 Amirov, M., 4, 209, 215 Barile, Margherita, 452 Anciaux, H., 450 Barinov, V.V., 569 Anderson, D.M., 443 Barnes, Ian S, 457 Andreevskii, D.V., 258 Barr, A.H., 333 Andrews, James, 581, 593 Barra, Mario, 132 Andrianov, I.V., 486 Barros, M., 581 Andrushkov, V.I., 44 Barry, C.D., 21, 30 Annaberdyev, E., 149 Barthel, Woldemar, 451, 459 Anpilogova, V.A., 77 Basilevich, I.A., 526 Apery, Francois, 418, 424 Bastatzkiy, B.N., 179, 182 Apraksina, T.I., 627 Basu, D., 83 Arbelais, Yuhanio Marulanda, 7 Baturin, A.I., 541 Archibald, R.C., xxix Bauer, Thomas, 675 Aribowo, A., 233 Bayata, Fatma, 62 Armenàkas, A.E., 74 Bazhenov, V.A., 57, 112, 116, 173, 382, 384, 385, 427, 445 Aronson, A.A., 263, 342 Beauville, A., 674 Artykbaev, A., 484 Behr, Richard, A., 145 Arvid, Perego, 660, 674 Beijing, 539 Aseev, V.I., 212, 413 Bekkar, M., 258 Aseev, V.V., 212, 413 Beleş Aurel, A., 624 Ashurbekov, K.D., 562 Belov, A.V., 62 Ashuri, K., 386 Belov, K.M., 74 Aumann, Günter, 329 Bergman, R.M., 46 Avdelas, A.V., 692 Bergmann, Horst, 691 Avdon’ev, E.A., 67, 219–221, 628, 648, 664, 667, 668, 670, 671 Bernstein, J., 429 Avinash, C. Singhal, 85 Bers, L., 450 Azevedo Tribuzy Renato de, 484 Bert, Jüttler, 655 Besyadovskiy, A.R., 219

S.N. Krivoshapko and V.N. Ivanov, Encyclopedia of Analytical Surfaces, 733 DOI 10.1007/978-3-319-11773-7, © Springer International Publishing Switzerland 2015 734 Name Index of the Authors of Additional Literature

Beyer, W.H., 614 Carter, J.S., 424 Bhattacharya, B., 4 Castro, Ildefonco, 474 Bîlă, N., 319, 320 Catalan, E., 77, 436 Biswas, Indranil, 692 Cayley, A., 166, 649 Bisztriczky, Tibor, 631, 674 Chan, T.K., 343 Blach, A., xiii Chapman, S.J., 418 Blachut, J., 131, 149 Charitos, C.H., 212 Blaga, P.A., 358 Chate, A., 145 Blank, YaP., 173 Chaykin, V.A., 145 Blaschke, W., 451, 459 Chegodaev, A.I., 173 Bleeker, D., 639 Chen, Bang-Yen, 317 Blinn James, F., 675 Chen, C.C., 449 Blotter, P.T., 145 Cheng, Hong-Yih, 257 Blum, Richard, 543 Cherdyntzev, N.V., 151 Bo, Pengbo, 3 Cherevatzkiy, V.B., 151, 157, 158 Bochillo, G.P., 315 Cheung, Y.K., 173 Böhme, R., 428 Chinenkov, YuV., 82, 486 Bodunov, A.K., 148 Chinenov, S.G., 58 Bodunov, N.A., 148 Choi, C.K., 82 Boem, W., 329 Chomilier, J., 258 Boersma, J., 4 Chong-Jun, Li, 329 Bogacki, S.l., xiii Christos T., Nakas, 520 Bolotin, V.V., 146–148 Chronowicz, A., 182 Bonet, E., 480 Chuaqui,M., 552 Bonnet, O., 544 Chubukina, L.P., 257 Bonetto, F., 480 Chumakov, G.A., 478, 482 Borisenko, A.A., 428, 561, 674 Chun, Changmook, 26 Bortovoy, V.V., xiii Churkin, G.M., 132, 514 Bottelli, S., 220 Cicala, Placido, 174 Bottema, O., 82 Clark, R.A., 113 Bourke, Paul, 280, 692 Coffman, A., 422, 654 Bousias Stathis, N., 229 Coghlan, Leslie, 422 Boy, W., 424 Cohn-Vossen, S., 48, 424 Boykov, I.K., 342, 355 Collet, P., 482 Bożek, B., 641 Collin, Pascal, 483 Boziev, I.A., 485 Coman, Gh., 328 Bracken, Paul, 475, 480 Cook, W.A., 145 Brakke, Ken, 457 Corno, M., 220 Brandner, G., 92 Courante, F., 445 Brandt, G.V., 142 Costa, A., 438 Brandt, Heinrich, 599 da Costa, Celso José, 450 Brankov, G., 501, 520 Craizer, M., 460 Brauner, H., 263, 546, 692 Cromwell Peter, R., 424 Brebbia, C.A., 183 Csonka, P., 174 Brehm, U., 424 Breiner, Ch., 429 Brent, Collins, 263 D Bright, Martin, 636 Dahman, W., 626 Brunella, Marco, 675 Dajczer, Marcos, 258, 484 Bubennikov, A.V., 79, 391, 568, 570, 593, 678 Dale, A.P., 329 Bulanov, G.S., 98 Dan, Dumitru, 433 Bulanov, S.N., 98 Dao Chong, Thi, 154, 454, 552, 553 Bulgakov, V.Ya., 32 Darevskiy, V.M., 125 Bunakov, V.A., 146, 148 Davis, R.F., 93 Burtzeva, S.V., 486 Dawoud, R.H., 628 Bushmelev, A.V., 418 Dedonder, Willy, 419 Buyske, Steven G., 453, 459 Degen Wendelin, L.F., 315, 329, 353 Degtyarev, V.M., 636 Dehtyar, A.S., 148, 173 C Delaunay, C., 156 Caballero, M., 581 Deniz Polat, B., 62 Caley, A., 00 Dennison, A.C., 182 Callahan, M., 455 Deriche, M., 128 Cardou, A., 226, 343 Dickson, S., 443 Carlson, K.H., 552 Dierkes, U., 460 do Carmo M., Dajczer M., 258 Dietrich, W., 229 Name Index of the Authors of Additional Literature 735

Dillen, F., 160 Fath el Bab, H., 483 Ding, Youdong, 328 Fauconnier, M., 82 Dipankar, Chakravorty, 85 Fediv, I.Ya., 453 Djashiashvilly, T.G., 342 Fedorenko, B.I., 98 Djondjorov P.A., 156 Fedorenko, N.A., 625 Diskovskiy, A.A., 486 Feldman, E.A., xxix Dmitrieva, N.P., 418, 692 Feng, R., 344 Dokula, S.M., 533 Fepl, Stanimir, 638, 692 Dooner David, B., 98 Ferrante, A.J., 183 Doǧan, Fatih, 358 Fetters, Lewis J., 439 Dong, S., 552 Fidrovskaya, N.N., 43 Dorfimeister, J., 484, 554 Filin, A.P., 481 Druzhinskiy, I.A., 93, 257, 536, 537 Filiz Ertem, Kaya, 54 Duan, H.L., 486 Finikov, S.P., 4 Duffty, B., 30 Firsanov, V.V., 145 Đukanović, Gordana, 649 Fischer-Colbrie, D., 450 Dumas, J.C., 358 Fleming, W.H., 343 Dupin, Ch., 343 Fleyman, N.P., 186 Dura, Gabriel, 258 Flöry, Simon, 73 Duren, P., 552 Flügge, W., 173 Dyhovichny, Yu.A., 164 Fogden, Andrew, 449, 457 Dzhandarbekova, D.D., 674 Fomenko, A.T., 94, 154, 419, 423, 428, 453, 474, 552, 553, 555 Dzhashiashvili, T.G., 260 Fomina, T.N., 428, 451 Dzhuraev, T.K., 459, 461 Fominskiy, L.P., 387 Forsyth, A.R., 559 Forrest, A.R., 329 E Francis George, K.A., 78, 422, 692 Earp, R., 474 Fraser, Ailana, 445 Edge, W.L., xviii Friaa, Ahmed, 182 Edlinger, R., 546 Friedman, R., 628, 674, 675 Edward J., Haug, 598 Friedrich, Th., 674 Eells, James, 156 Frohman, Charles, 457 Efimov, M.I., 569 Fujioka, A., 460, 544 Efimov, N.V., 614 Furuhata, Hitoshi, 460 Efstifeev, M.F., 263 Eisenhart, L.P., 559 Eisland, J., 664, 669 G Eklundh, J-O., 186 Gackstatter, Fritz, 449, 450 Emel’yanova, E.M., 521 Gadzhimuradov, M.A., 678 Emel’yanova, Yu.V., 150 Gaffet, B., 636 Emin, Kasap, 74 Gagarin, Yu.A., 157 Epstein, Beryl, 263 Gaidalovich, V.G., 482 Epstein, H., 482 Gaidar, O.G., 433 Epstein, Sam, 263 Galeta, E.A., 567 Eremeev, V.V., 674 Galimov, D., 172 Eremeeva, L.N., 625 Galimov, Sh.K., 265, 485 Eremin, V.D., 486 Gallavotti, G., 482 Erhalev, V.I., 481 Gallego, Francisco Javier, 74 Ermakov, A.V., 537 Galletly, G.D., 343 Escher, M.K., 417 Gamdy, Paul J.F., 457 Ettlinger, Hyman Joseph, 655 Ganeeva, M.S., 144 Eugene, Fiume, 2 Gansca, I., 328 Euler, Leonard, 4, 429 Garib, M.G., 173 Garland, M., 552 Garstecki, P., 439 F Gavelya, S.P., 257 Fam Hong, Mga, 190 Gaydaychuk, V.V., 57, 112, 116, 173, 382, 384, 385, 427, 445 Fan, Da-jun, 486 Gayupov, G.N., 484 Fan, S.C., 128, 173 Gentile, G., 480 Fares, M.G., 191 Gershon, Elberb, 74 Fardis Michael, N., 229 Getzevich, E.K., 418 Farin, G., 329 Gfrerrer, A., 229 Farshad, M., 145, 148 Ghosh, K.K., 83 736 Name Index of the Authors of Additional Literature

Giacomo, Frulla, 537 Hagen, H., 428 Gielis, J., 539 Haglund, Frédéric, 678 Gînsča, I., 93 Hahn, L., 86 Giraldo, Luis, 74 Haizhong, Li, 544 Girard, J.P., 358 Halabi, S.M., 76, 79, 137, 169 Girloy, T.I., 113 Hamdoon, F., 263 Girsh, A.G., 49 Hano, Jun-ich, 156 Glaeser, Georg, 2, 80 Hansford, D., 329 Glässner, Ekkehart, 459 Harbourne, Brian, 675 Glogovskiy, V.V., 685–687 Harper, Paul E., 439 Gmainer, J., 166 Hartshorne, R., 628 Gobel, M., 423 Hasanis, Thomas, 450 Goemans, W., 160, 174 Haubitz, Imme, 451, 459 Gogoberidze, Ya.A., 179, 182 Hauser, H., 141 Goldenweizer, A.L., 43, 46 Haustova, N.D., 93 Goodchild, M.F., 214 Hauswirth, Laurent, 483 Gorbatovich, J.N., 14, 46 Havlicek, Hans, 166 Gordienko, V.M., 482 Hayamizu, Y., 233 Gorjanc, Sonja, 83, 650, 653, 655, 656 Hee-Seok, Heoa, 74 Gorodov, G.F., 157 Hen Kye, J., 145 Gotman, A.Sh., 220 Henkee, C.S., 443 Gotzulyak, E.A., 57, 112, 116, 173 Henneberg, L., 441 Gould, Ph.L., 145 Henshall, J.L., 257, 343 Goursat, E., 325, 642–644 Hilbert, D., 48, 417 Goziridze, A.F., 428 Hildebrandt, S., 445, 460 Grabko, S.M., 211 Hill, J.E., 665 Grant, K., 472 Hilmi Hacısalihoglu, H., 54 Gray, Alfred, 73, 89, 240, 390, 417, 420 Hirashima, Masaharu, 431 Grechishnikov, V.A., 257 Hismatulin, N.I., 265, 485 Greer, J.M. Jr., 343 Hirose, T., 233 Gribkov, I.V., 481 Hirz, M., 229 Gridgeman, N.T., 333 Hitt Richard, L., 258 Grigir’ev, A.M., 157 Hoang Zuy Thang, 659 Grigorenko, Ya.M., 300, 311, 386, 486 Hoffman, D., 438, 443, 445, 449, 455 Grigor'yev, S.V., 258 Hohenberg, Fritz, 678 Grischak, V.Z., 62 Holgate, Th.F., 649 Gritskevich, O.V., 115 Holyst, R., 439 Gritzishen, I.V., 357 Honeker, Ch.C., 439 Gromov, M.Ya., 419 Hopf, Helinz, 472, 482, 483 Groβe-Brauckmann, K., 439, 483 Horoshun, L.P., 145 Grozdana, Radivojević, 82 Horst, Martini, 263 Gruber, Franz, 2 Horvath, I., 179, 183 Grünbaum, B., 678 Hoschek, J., 546 Gruner, Sol M., 489 Hu Hesheng, 482 Gruter, M., 445 Hu, Shi-Min, 21 Gryanik, M.V., 520 Huang, C., 93 Gu Chaohao, 482 Huang, Wu-Hsiung, 484 Guggenberger, W., 145 Hudson, R., 646 Guid, Niko, 328 Huili Liu, 460 Gulyaev, V.I., 57, 112, 116, 151, 173, 260, 382, 384–386, 398, 427, Huisman, Johannes, 692 445 Hunt, Bruce, xviii, 661, 675 Gumen, N.S., xiiv, 649, 689 Husty, M., 166, 608 Günter, A., 329 Hvylya, I.K., 521 Gurevich, I.I., 77 Hyeng, Christian A. Bock, 233, 340, 343, 362, 385, 486, 509, 531 Gurevich, K.Yu., 674 Gurevich, V.I., 77, 151 Gustavo, Gordillo, 134 Guy, R.K., 646 I Guzul, I.S., 481 Iancău, V., 93 Ibolya Szilágyi, D.M., 635 Ignatenko, V.F., 664, 678 H Il’hamov, U., 484 Hadid Hassoun, A., 179, 182, 183 Iliev, V., 692 Hadzhilazova, MTs., 156 Inoguchi, J., 544 Haeberlein, M., 449, 457 Ionin, V.K., 475, 688 Haeberli, Paul, 4 Isenberg, Cyril, 428 Name Index of the Authors of Additional Literature 737

Ishihara, Toru, 418 Kirov, V.S., 481 Istotskii, V.V., 258 Kiselev, A.V., 260 Ito, Miori, 5 Klamkin, Murray, 74 Iura, Masashi, 431 Klepikova, L.S., 418 Ivanov, G.S., 592, 669, 674 Klimanov, V.I., 62 Ivanov, V., 226 Klimenko, S., 423 Ivanov, V.N., 19, 71, 138, 174, 186, 194, 199, 206, 242, 243, 267, 270, Klinowski, Jacek, 439, 457 273, 297, 303, 313, 340, 342-344, 346, 349, 350, 352, 355, 357, Klyachin, V.A., 153 365, 367, 378, 380, 381, 385, 395, 486, 493, 502, 504, 523, 548, Knabel, J., 74, 257 550, 559 Knutsen Andreas Leopold, 675 Ivanova, S.Yu, 149 Kobayashi, R., 453 Ivey, Thomas, 315 Kobayashi, S.P., 442 Ivont’ev, A.V., 263 Kobko, V.P., 208 Kochetkova, A.L., 212, 628 Kogan, B.Yu., 93 J Koh, V.N., 567 Jackman, Henrik, 588 Koiso, Miyuki, 156 Jackway, P.T., 128 Kokovin, N.I., 214 Jae-Hoon, Kanga, 145 Kolchunov, V.I., 485 Jaiswal, O.R., 131 Kolesnikov, A.M., 151 Jakomin, M., 164 Kolesov, N.V., 258 Jankowski, Jacek, 62 Kolomak, V.D., xviii Jank, Walter, 73, 74 Kolomiets, E.A., 500 Jeff, Bryant, 140 Komijany, Y., 399 Jessop, C.M., 636 Kompanietz, L.A., 674 Jin Gon, Kim, 145 Kondrashov, A.N., 173 Joachimsthal, F., 557 Kopytko, M.F., 34, 37, 206 Jolicoeur, C., 226, 343 Korjakin, A., 145 Johnston, M.E., 482 Korneva, I.P., 212 Jorge, L.P.M., 450 Kornienko, A.V., 390 Juhász, I., 329 Kornienko, L.Z., 212 Jung, Y., 457 Korn, G., 614 Jürgen, Köller, 398 Korn, T., 614 Kornishin, M.C., 206 Korobov, L.A., 486 K Korolev, E.A., 325, 428, 451 Käferböck, Florian, 460 Korotich, A.V., 73 Kairov, A.S., 145 Korsakova, L.G., 212 Kalinin, V.S., 151 Kosel, F., 164 Kalra, M., 520 Kosel, T., 164 Kamanin, L.N., 97 Koslov, A.T., 624 Kantor, B.Ya., 63, 257 Kosolapova, L.A., 144 Kantor, B.E., 561, 565 Kostyrenko, V.V., 145 Karagashev, D.A., 260, 342 Kotaro, Yamada, 441 Karcher, H., 445, 449, 455 Kotov, I.I., 208, 214, 323 Kardashevskaya, Yu.G., 9, 14 Koutrouﬁotis, Dimitri, 450 Karpov, V.V., 145 Kovalev, S.N., 263 Kaschenko, A.V., 263, 505, 506, 520 Koval’chuk, P.S., 145 Kashafutdinov, S.T., 537 Koval’skiy, B.S., 43 Kashina, I.V., 74, 575 Kovalenko, I.B., 459 Kasue, Atsushi, 552 Kovnzsov, N.I., 449, 481 Katsuhiro, Moriya, 454 Kozhevnikov, A.Yu, 265 Kawahara, K., 233 Kozlov, S.V., 145 Kawakami, Y., 453 Krames, Josef, 561, 631, 650, 675, 656, 674 Kenmotsu, K., 156, 484 Krapivina, G.I., 417 Kharabayev, Yu.N., 74, 569, 592, 607, 608, 611, 612 Krasic, Sonja, 54 Kheyfets, A.L., 172 Krasnov, V.A., 674 Kiehn, R.M., 206 Krauter, Peter, 459 Kienzle, Otto, 4 Kreychman, M.M., 151, 157 Kiesling, Ernst W., 145 Krikanov, A.A., 144 Kilian, M., 442 Krishna Reddy, G.V., 342 Kim Gia, 439 Krivoshapko, S.N., 2–6, 10, 11, 15, 18, 22–24, 26–28, 30, 31, 34, 42, Kifer, Yuri, 552 65, 69, 76, 79, 83, 86, 87, 106, 111, 118, 133, 137, 140, 144, Kircher, S., 552, 568 149, 150, 169, 174, 180, 181, 206, 212, 229, 233, 300, 303, 340, Kirilov, S.V., 92, 218, 592 343, 348, 357, 358, 362, 363, 482, 486, 501, 508, 509, 514, 521, Kirischiev, R.I., 13, 36, 278 527, 531– 533, 569, 592, 623 738 Name Index of the Authors of Additional Literature

Kruglyakova, V.I., 343 Lockwood, E.H., 406 Kruppa, E., 546 Logan, A., 649 Krus, Petter, 537 Loman, V.I., 520 Krutov, A.V., 4, 34, 37, 108 Looijenga, E.J.N., 626, 628 Kruzelecki, J., 149 Loop, Ch.T., 329 Krylov, I.P., 636, 665 López, Francisco J., 450, 455 Kubenko, V.D., 145 López, Rafael, 484 Kudenko, S.M., 625 Lu, S.N., 544 Kuen, T., 481 Luah, M.H., 128 Kühnel, W., 472 van Luijk, R., 649 Kukudjanov, S.N., 62 Lushin, V.N., 537 Kummer, E., 646 Lusta, G.I., 569, 575 Kupar, A.K., 617 Lynn, Paul P., 179, 182 Kupriyanova, G.Ya., 212 Lytkina, E.M., 688 Kuramin, V.P., 66 Lyukshin, V.C., 231-233, 235, 239 Küronya, Alex, 675 Kushnarenko, I.V., 345, 347 Kusner, Rob., 483 M Kustch, N.V., 74 Maan, H. Jawad, 62, 145, 619 Kutzenko, G.V., 112 Makarova, K.P., 481 Kuzeev, R.R., 475 MacDonnel, Josef, 418 Kuzenko, L.M., 542 McCarthy, J.M., 92 Kuzmenko, E.A., 6 Maeder, R., 437 Kuzyutin, V.F., 674 Maehata, Tatumi, 44, 164, 174 Kvaskovm, B.I., 328 Maevskiy, E.V., 559 Mahmud Hussain, Al-Hadh, 342 Mamford, D., 669 L Mandelbaum, Richard, 674 Lagutin, S.A., 95 Makarov, A.I., 62 Lai, Yu-Kun, 21 Makarov, G.I., 173 Lakirev, S.G., 58 Makarova, K.P., 481 Lalan, V., 544 Makeev, E.B., 149 Landsmann, G., 344 Makowski, J., 145 Lang, J., 229 Maleček K., 2, 96, 343 Lantuh, L.G., 173 Maleev, M.V., 62 Lawson, N.V., 427 Maleshich, Y., 424 Lebedev, V.A., 520, 521 Mallett, R.L., 258 Lee, I.K., 263, 568 Maltzeva, L.M., 257 Lee, Jay, 214 Mamay, V.I., 144 Leissa, Arthur W., 145 Mamford, D., 669 Lemaire, L., 554 Mamieva, I.A., 42, 65, 83, 87, 149, 521 Lenza, Pietro, 174 Mammana, Carmelo, 212 Leont’ev, N.N., 173 Mamuzok, G.A., 373 Leopoldseder, S., 4, 226 Manasherov, E.E., 74 Le Tuan, K., 258 Mandelbaum, Richard, 674 Lewiner, T., 450, 460 Manjunath, K.S., 257 Lewinski, T., 74, 257 Mann, Stephen, 2, 73 Leys, Jos, 692 Manselli, P., 406 Li, C., 539 Mansﬁeld, E., 42 Li, F., 539 Manuel Munoz Saiz, 537 Li, W.Y., 173 Marar, W.L., 424 Lichtenfels, O., 442 Marcel, Gagnon, 85 Lidin, S., 449 Marković, Miroslav, 674 Lidskiy, V.B., 43, 46 Markarov, S.M., XV Likar, Matjaž, 328 Markelov, N.A., 211 Lipkin, H., 95 Martín Francisco, 450 Lipnitzkiy, M.E., 521 Martin Man-Chun, Li, 445 Lisenkov, V.T., 669 Martin, Ph., 358 Lisichkin, S.E., 263 Martiner, A., 475 Lisitsa, V.T., 258 Martinez, A., 472 Lisowski, A., 85 Martirosov, A.L., 32, 568, 569, 588 Liu, Dingyuan, 329. Martynenko, M.D., 173, 190 Liu, Yong-Jin, 21 Maryukova, M.E., 481 Liu, Zhaoshgne Song Zuozong, 688 Maryukova, N.E., 475 Ljubica, S. Velimirović, 82 Masaaki, Umehara, 441 Name Index of the Authors of Additional Literature 739

Masal’tsev, L.A., 559 Montealegre, Christina R., 474 Masato, Ito, 429 Monterde, J., 329 Masatoshi, Kokubu, 441 Montiel, Sebastián, 484 Mason, D.R., 145 Morgan, J.W., 628, 675 Mastropierto, V., 480 Moriya, K., 454 Mathieu, Gil-oulbe, 86, 181, 342, 348, 482, 559 Mornon, J.P., 258 Matveev, V.S., 418, 423 Moroz, S.V., 173, 190 Mbakogu, F.C., 168, 173 Morozov, A.P., 173 Mcbride, W., 257 Mota Soares, C.A., 148 McCarthy, J.M., 92 Mota Soares, C.M., 148 McIntosh, J., 554 Muha, I.S., 206, 342 Mckinnon, D., 649 Muhadze, L.G., 428 Mednykh, A.D., 682 Muhin, O.I., 692 Medvedev, V.I., 263 Müller, E., 650, 656 Meeks III, W.H., 438, 450, 455 Müller-Stach, Stefan, 675 Mehta Kishor, C., 145 Mumford, D., 674 Meinicke, Eb., 73 Münch Mechthild, 46 Meirer, Klaus, 74 Munteanu, M.I., 174 Melin, Tomas, 537 Muradov, Sh., 212 Mellerovich, G.M., 63 Murat Kemal Karacan, 358 Melnik, V.I., 208, 212 Murre, J.P., 674 Mendonca, P.R.S., 149 Myung-Soo Kima, 74 Meng Sun, 2 Merkin, D.R., 47 Meseguer, F., 486 N Meshcheryakov, N.A., 115 Nabil S. Hadawi, 173 Meshkov, V., 455 Naboulsi, S.K., 343 Meusnier, J.B., 429 Naeve, A., 186 Micale Biagio, 212 Nankov, G., 226 Mića S. Stanković, 82 Nartova, L.G., 459, 461 Mick Sybille, 674 Narzullaev, S.A., 212, 304 Miftahutdinov, I.H., 428, 457 Nasrulaev, F.S., 561, 562 Mihailescu, M., 179, 183 Nassar H. Abdel-All, 482 Mihaylov, N.P., 484 Naumenko, V.V., 63 Mikhailenko, V.E., 32, 209, 211, 212, 215, 263, 486, 521, 674 Naumovich, N.V., 212 Miklyukov, V.M., 358 Nayak, A.N., 83 Milán, F., 472 Naydysh, V.M., 212 Mileykovskiy, I.E., 173, 617 Nayhanov, V.V., 212 Miller, J.R., 206, 626 Nazarov, G.I., 125, 126 Milnor, J., 641 Nedeshev, Yu.B., 125 Milousheva, V., 581 Nemirovskiy,Yu.V., 257 Minaev, L.S., 173 Neovius, E.R., 435 Minakawa Couichi, 44, 164, 174 Netzvetaev, N.Yu., 628 Minakawa Youichi, 174 Niče, Vilko, XVIII, 631 Mintz, I.Ya., 263 Nikitenko, O.P., 678 Mintz, L.I., 263 Nikitin, A.P., 145 Mironkov, B.A., 173 Nikitin, I., 423 Miroshnichenko, A.V., 263, 373 Nikityuk, V.A., 333, 336, 337 Miroshnichenko, V.L., 328 Nikolaev, A.P., 144 Mirza, J.F., 521 Nikolaevskiy, G.K., 55 Mischenko, A.S., 94, 423, 474 Nikolaevskiy, Yu.A., 674 Misikov, B.R., 482 Nikonorov, S.V., 481 Mitenkov, F.M., 157 Nistor, A.I., 174 Mitrofanova, S.A., 569 Nitshe Johannes, G.C., 433, 445 Mityukov, M.M., 144 Nomizu, Katsumi, 156, 166, 459 Miyaoka, R., 453 Nordstrand, T., 288, 422, 481, 642, 661, 681 Mladenov, I.M., 156, 157 Novikov, S.P., 552 Mlodzievskiy, B.K., 325 Mochernyuk, N.T., 209 Mohajerzadeh, S., 399 O Mohd Altab Hossain, 474 Oancea, Nicolae, 258 Moiseeva, V.E., 144 Obradović, Marija, 649 Moishezon, Boris, 674 Obraztzov, I.F., 62, 146, 148 Molenaar, J., 4 Obukhova (Obuhova), V.S., 2, 11, 32, 33, 73, 74, 206, 212, 486, 521, Mollin, R.A., 626 669, 674 Monge, G., 4, 206 Odehnal, Boris, 353 740 Name Index of the Authors of Additional Literature

Oetter, R., 30 Pollicott, Mark, 482 Ogawa, A., 441, 442 Polthier, Konrad, 449, 455 Ogloblya, A.I., 263 Polyakova, E.V., 145 Olejníková, T., 258 Popa, Emil M., 319 Olmsted, J.M.H., 626 Popa, Ionuţ, 258 Onanov, G.G., 62 Popov, A.G., 62, 109, 482 Osgood, B., 552 Popov, A.Yu., 125 Onischuk, N.M., 186, 459, 692 Popova, L.S., 674 Ortega, M., 581 Postnikov, M.M., XV, 00 Osserman, R.A., 428, 432 Potapov, S.Yu., 387 Ostianu, N.M., 319 Potapov, Yu.S., 387 Özgül, Keleş, 62 Pottmann, Helmut, 4, 73, 91, 226, 329, 460, 568 Poupon, A., 258 Poznyak, E.G., 474, 480, 482 P Preobraztenskiy, I.N., 62 Pabishek, E., 74 Prohorenko, F.F., 144 Palazotto, A.N., 343 Proskurenko, D.A., 263 Pallagi, J., 688 Protasov, V.D., 146-148 Palm, G., 128 Protas'ev, V.B., 258 Palman, Dominik, 343, 692 Protod’yakonov, S.M., 67, 219-221, 628, 648, 664, 667, 668, 670 Pamin, J., 74 Prusakov, A.P., 486 Panasyuk, K.S., 236 Pucci, C., 406 Panina, G., 561 Puchkov, A.A., 125, 126 Panov, O.V., 55 Pul’pinskiy, Ya.S., 153-155, 158 Panzani, G., 220 Pushkar, Joshi, 631 Papantoniou, B.J., 212 Putinar, Mihai, 319 Parinov, P.M., 481 Park, F.C., 26 Parker, J., 682 Q Parshin, A.N., 674 Qatu, M.S., 145 Patlashenko, I.Yu., 145 Patterson, T., 95 Paukowitsch, Hans Peter, 206 R Pavilaynen, V.Ya., 63, 485 Rabinovich, A.I., 486 Pavlovic, M.N., 168, 173 Rachkovskaya, G.S., 74, 568, 569, 577, 592, 607, 608, 611, 612 Pedit, F., 554 Rachkovskaya, N.S., 74, 592, 607, 608 Pelipenko, V.V., 481 Radwanska, M., 74 Peng, C.K., 544 Radzevich, St.P., 574, 581, 593 Perchanik, N.E., 678 Raffy, L., 559 Perel’man, G.Ya., 561 Ragni, Piene, 655 Perez, A., 92 Raizer, V.D., 74 Peternell, Martin, 4, 344 Ramaswamy, G.S., 145, 160 Peterson, K.M., 325, 326 Ramiro-Manzano, F., 486 Petit, Jean-Pierre, 418 Randrup, T., 263, 568 Petropavlovskaya, I.A., 74 Rashevskiy, P.K., 475 Petrovskiy, I., 674 Rasskazov, A.O., 74 Petrova, A.T., 263 Rath, Wolfgang, 674 Petukhova, G.I., 6 Reckziegel, H., 481 Pianowski, Lesław, 60 Reichhart, Adam, 486 Pichkov, S.N., 157 Reid, M., 633 Piero, Gil, 537 Reissner, E., 113, 145, 431 Pilarski, I., 82 Rekach (Rekatch), V.G., 2, 22, 164, 176, 206, 357, 358, 363, 386-388, Pilipaka (Pylipaka), S.F., 10, 33, 79, 206, 231, 263, 297, 559 431, 476 Pikul, V.V., 621 Rembs, E., 478 Pinkall, Ulrich, 166, 449 Ren-Hong Wang, 329 Plateau, J., 427 Renton, J.D., 62 Pleshakov, V.F., 76 Repovsh, D., 424 Podarov, K.A., 63 Reva, V.G., 542 Podgorniy, A.L., 2, 32, 73, 212, 486, 521, 659 Reznichenko, K.A., 257 Pod'yapol'skii, Yu.V., 115 Rikards, R., 145 Pogorelov, A.V., 46, 427, 450, 459, 484 Rimrott, F.P.J., 128 Pogrebezskaya, M.N., 235 Rizwan, Muhammad, 186, 194, 199, 202, 206 Pokidyshev, G.S., 649 Roberts, A.W., 257 Polański, Stanislaw, 60 Robinson, M., 145 Polat, B. Deniz, 62 Rodriguez, I., 486 Polivanov, A.A., 62 Roeschel, Otto, 417 Name Index of the Authors of Additional Literature 741

Rogers, C., 482 Seiler, W., 482 Romanova, V.A., 521 Semenov, A.A., 145 Rong-Shine Lin, 74 Semkina, E.E., 451 Ros, A., 455 Séquin, Carlo H., 581, 593, 631 DeRose, Tony D., 329 Serdyuk, V.E., 418 Rosenberg, Harold, 445, 450, 475, 483 Sevruk, V.N., 233 Rosendorn, E.R., 563 Shahabadi, M., 399 Rossman, W., 441, 442 Shalamov, V.G., 257 Roth, A., 329 Shambina, S.L., 34, 180, 212, 569, 592 Rottmann, H., 263 Shangina, E.I., 678 Roulleau, Xavier, 675 Shapiro, A.V., 173 Roussos, Jaonnis M., 258, 544 Sharp, John, 263 Rozhok, L.S., 486 Shavel, I.H., 675 Rubin, O.D., 263 Shefel, S.Z., 561 Rudman, L.I., 592 Shein, V.T., 175, 209 Rudnitzkiy, O.I., 674 Sheveleva, G.I., 263 Rybakov, V.N., 74 Shibata, Choko, 475 Rynkovskaya, M.I., 431 Shikin, E.V., 474, 480, 482 Ryzhkov, N.N., 18, 21, 28 Shinkarenko, G.A., 342 Ryzhkova, T.G., 257 Shleykov, I., 172 Shoichi, Fujimori, 441 Sholomov, I.H., 567, 574, 592 S Shoman, O.V., 686 Sa Earp, Ricardo, 472 Shrivastava, N.K., 257 Sabitov, I.H., 485, 682 Shtaerman, Yu.Ya., 179, 182 Sachenkov, A.A., 62 Shubnikov, V.G., 674 Sachs, Hans, 74, 232, 546 Shulikovskii, V.I., 166, 260, 386, 390, 432, 472, 559 Sadovnichiy, A.F., 173 Shvidenko, Yu.Z., 236 Sahabutdinov, A.G., 342 Šibrava, Zd., 343 Sakao, Keishi, 174 Sievert H., 479 Salamon, Simon, 73 Silaenkov, A.N., 74, 674 Salkowski, E., 460 Silhol, Robert, 675 Samartin, A., 173 da Silveira, A., 474 Sameen, Ahmed Khan, 626 Simenko, O.V., 651 Sanjeev, Bedi, 2, 73 Simitses, J.G., 39 Sankin, Yu.N., 486 Simmonds, J.G., 235 Santaló, L.A., 480 Sinchenko, L.D., 212 Sasaki, Takeshi, 459, 460 Singatullin, R.S., 475 Sauvigny, F., 460 Sinitzin, A.V., 149 Savaresi, S.M., 220 Sinitzina, O.V., 674 Savelov, A.A., 50 ŠipušMŽ.., 160 Savula, Ya.G., 34, 37, 186, 206, 342 Sitnikov, V.S., 55 Schaal, Hermann, 451, 460, 546 Sivak, V.F., 145 Schegol’kov, N.I., 257 Skaggs, Robert L., 22 Scherk, H.F., 432, 443 Skidan, I.A., 190, 197, 206, 344, 499, 500, 568, 569 Schicho, Josef, 60, 344 Skouteropoulou, Anna-Maria O., 229 Schief, W.K., 482 Slav, L.I., 263 Schmitt, N., 442 Smerichko, O.V., 689 Schoen, A.H., 439, 455 Smirnov, S., 418 Schoenberg, I.J., 328 Smirnov, V.I., 626 Schrott, Michael, 353 Smith, T.A., 343 Schüffer, Karjheinz, 483 Soare, M.V., 85, 86, 624 Schultz, B., 688 Soare, Nicolae, 460 Schulz, Ch., 418 Soh, C.K., 343 Schurov, I.A., 257 Sokolov, V.V., 263, 342 Schwalbe, D., 450 Soldatos, K.P., 39, 46 Schwartz, A., 422, 434, 655 Solov’yov, Yu.P., 94, 474 Schwarz, H.A., 434 Sols, Ignacio, 74 Sdvizhkov, O.A., 206 Song, J.L., 263 Seaman, Walter, 483 Souriau, Jérôme, 418 Sedaghat, Pisheh H., 399 Sovertkov, P.I., 482 Sedletzkaya, N.I., 209 Spelta, C., 220 Sedov, V., 263 Środa, P., 641 von Seggern, D.C.R.C., 284, 529 Stanislavskaya, L.A., 678 742 Name Index of the Authors of Additional Literature

Stanton, C., 422, 655 Toﬁl, Jolanta, 74, 77 Stasenko, I.V., 342, 366 Togliatti, E.G., 656 Stavridis, L.T., 74 Togino, Kadzuto, 692 Stefanova, St., 554, 555 Tölke, Jürgen, XVIII, 74, 546 Steller, M., 472 Tomarevskaya, E.S., 55 Stepanov, S.E., 481 Tomasz, Kubiak, 62 Sterling, I., 442, 483 Tomi, F., 445 Stolyarchul V.A., 148 Tompson, J.J., 343 Strubecker, Karl, 287, 432, 459 Torquato, S., 457 Struwe, M., 445 Toubiana, Eric, 441, 450, 472, 474 Stumpf, H., 145 Tovstik, O.E., 145 Stupishin, L.Yu., 149 Tovstik, P.E., 43, 46 Stylianos, S. Stamatakis, 546 Tramberend, H., 423 Sultanov, B.Z., 137 Trifanov, A.E., 486 Sulyukmanov, F.C., 263, 288 Troﬁmov, V.N., 552 Sun Bo-Hua, 128 Tromba, A., 428 Sun G.J., 344 Trukhina, V.D., 74 Suresh, G.R., 145 Trushin, S.I., 7 Surkova, G.I., 373 Tuhman, Ya.P., 625 Suslov, G.K., 95 Tuzhilin, A.A., 428 Švec, Alois, 317 Tzeitlin, A.A., 485 Svetlanova, O.I., 186 Tzitzéica, Gheorghe, 119, 319, 320, 323 Svistov, A.Ya., 212 Tzvetkova, E.G., 149 Sir Peter Swinnerton-Dyer, 636 Svoboda, Karel, 482 Sysoev, Yu.A., 257 U Szarkova, Dagmar, 2, 96 Udrişte, C., —321">319—321, 323 Szefer, G., 85 Umehara, M., 474, 483 Szemberg, Tomasz, 675 Urgen, Mustafa, 62 Szirmai, J., 688 Ushakov, V.V., 674 Utishev, E.G., 505, 506 Utkin, G.A., 649 T Uzakov, H., 211 Taha, Youssef, 482 Takehiro, Iton, 317 Taku, Sato, 429 V Talenti, Giorgio, 552 Vagner, V.V., XVIII, 00 Tambuleá, L., 328 Valcea, S.A., 320 Tanelli, M., 220 Van-Brunt, B., 472 Tanner, John L., 173 Vanĕk, Jiří, 82 Tarasov, A.G., 65 Varlamov, V.I., 260 Tarasov, V.M., 541 Varshavskiy, I.P., 65 Tarnai, Tibor, 44, 173 Varvak, M,Sh., 173 Taubin, G., 660 Varvaritza, A.G., 35, 214 Teixeira, R., 460 Vasilenko, O.V., 173 Templer, R.G., 5 Vasil’ev, A.N., 115 Tenca, Luigi, 4 Vasil’ev, O.B., 542 Teng, J.G., 62, 145 Vasil’ev, V.V., 146-148 Teodor Virgil, 258 Vasil’eva, O.V., 153 Teraoka, Atsuo, 257 Vasil’kov, B.S., 173, 485 Teslya, V.A., 173 Vassilev, V.M., 156 Tevlin, A.M., 212, 263, 288 Vaynberg, D.V., 260, 398 Tham, L.G., 173 Velichová, Daniela, 343 Thanon, A.Y., 183 Verhovskiy, A.B., 257 Thomas, Edwin L., 443 Verstraelen, L., 263, 460 Thomsen, G., 460 Vesnin, A.Yu., 682 Ticlete, G., 93 Virich, S.O., 263, 559 Tigaryev, V.M., 226 Vladimorov, V.S., 552 Timan, A.F., 552 Vlasov, V.G., 485 Timonin, A.M., 300, 311 Vogel, W., 212 Timoshin, M.A., 35 Volkmer, Reinhard, 451, 459 Tishkov, V.V., 145 Volkov, G.F., 160, 176, 177 Tkachev, V.G., 358 Voloshin, E.I., 428 Tkacheva, V.A., 153 Vorobkevich, R.I., 11 Toda Magdalena, 482 Voronchihin, M.A., 417 Todd, A. Alonzo, 520 Voronina, A.N., 15, 16 Name Index of the Authors of Additional Literature 743

Voros, A., 482 Y Voss, K., 482 Yadgarov, D.Ya., 567, 568, 570, 574, 581, 585, 586, 592, 598 Vranceanu, G., 319 Yakovlev, V.A., 568, 592 Vrancken, L., 460 Yakubovsky, A.M., 342 Vyzantiadou, M.A., 692 Yakupov, N.M., 206, 257, 265, 485 Yamada, K., 474, 483 Yamamoto, K., 233 W Yamanoi, K., 453 Wagon, S., 450 Yan Hong-Sen, 257 Wahl, J., 626, 628 Yanhua Yu, 460 Wake, G.C., 484 Yang, T.Y., 173 Wallner, J., 4, 73, 91, 226, 568 Yao, K., 539 Wan, F.Y.M., 257 Yao Xinghui, 688 Wan, F., 431 Yarahmedov, G.A., 561 Wang Guo-Zhao, 552 Yarin, L.I., 521 Wang Wenping, 3 Yasuzawa, V., 145 Wanga, Y., 486 Yayli, H., 54 Waszczyszyn, Z., 74 Yayli, Yusuf, 358 Wayne, Rossman, 441, 442 Yeh Kai-Yuan, 128 Wei, F.S., 449 Yeom, D.J., 145 Weihuan, Chen, 544 Yilmaz, Tuncer, 358 Weiner, Joel L., XIX, 552 Yong-Xia Hao, 329 Weingarten, J., 472 Yoon Young Kim, 145 Weiss, Gunter, 263 Yoram, Koren, 74 Weiβ,Günter, 73, 74, 82, 674 Young, L.C., 552 Weissmüller, J., 486 Yu Junghyun, 26 Weisstein, Eric W., 90, 121, 138, 217, 240, 283, 438, 441-443, 449, Yu Kawakami, 441 478, 479, 481, 619, 627, 628, 638, 657, 660, 680 Yu, S.K., 343 Wells, D., 132 Yurasov, S.Yu., 263 Welter, J., 30 Wen-Guang Jiang. 257, 343 Wenninger, M., 678 Z Wente, Henry C., 442, 483, 559 Zaﬁropoulos, S., 692 Werner, D., 482 Zalgaller, V.A., 153, 484 Werner, A.L., 561, 565 Žalik, Borut, 328 Whathan, J.F., 343, 358 Zamyatin, A.V., 74, 575 Whiston, G.S., 343 Zarutskiy, V.A., 145 Whittemore, J.K., 260 Zavriev, K.S., 428 Williams, Orlan G., 22 Zav’yalov,Yu.C., 328 Wills, J.M., 418 Zenkevich, N.A., 674 Wilson, L., 639 Zenzri, Hatem, 182 Winkler, F., 344 Zeyliger, D.N., 2 van de Woestyne, I., 160, 174 Zhang, H., 149 Wohlrab, Or., 475 Zhang, Wei, 128 Wolf, G.W., 214 Zharikova, L.A., 212 Wong, K.K.Y., 149 Zhou Zixiang, 482 Wu, H., 484, 554 Zhukovskiy, E.Z., 164 Wunderlich, W., 4, 12, 13, 15, 36, 418, 678 Zhukovskiy, N.E., 95 Zhulaev, V.P., 137 Zmijewski, Krzysztof Henryk, 624 X Znamenskiy, D.I., 258 Xah, Lee, 263 Zolotuhin, V.F., 567, 568, 592 Xenos, Ph J., 212 Zolotuhin, Yu.P., 418 Xie, Jiang, 169 Zubritzkaya, M.A., 263, 342 Xu, Gang, 552 Zubov, L.M., 257 Xu, Wei, 328 Xu, Z., 344 Surfaces

A the B surfaces, XVII Aerodynamic surface given by a continuous framework of Blutel surfaces, 315 elliptical ribs, 219 Bohemian dome, 389 water-lines in the form of generalized Agnesi curls, 219 Bonnet surfaces, 543 Aerodynamic surfaces, 219 Bour’s minimal surface, 436, 446 Aerodynamic surfaces given by algebraic plane curves, 219 Boy surface, 423 Afﬁne minimal surfaces, 451, 459 “Bracelet”, 414 Afﬁne sphere of elliptical type, 321 Bress hyperboloid, 691 Algebraic minimal surfaces, 453 Bullet nose, 126 Algebraic quartic surface, 636 Algebraic quantic surfaces, 656 Algebraic surfaces of the C 8th order, 670 Canal surfaces, 339, 343 high orders, 627, 628 of Joachimsthal, 284, 344, 559 seventh order, 223, 665 of Joachimsthal in the lines of principle curvature, 346 third order, 322, 353, 355, 631 Cartan umbrella, 78 12th order, 672 Carved Analytical topographic surfaces, 214 sinusoidal surface, 200 Anticlastic surface, 474 surface, 185, 207 Antisphere, 108 surface with directrix cubic parabola and generating ellipse, 200 Arch surface, 233 surface with directrix cycloid and generatrix ellipse, 201 Archimedean serpentine, 233 surface with directrix ellipse and generatrix parabola, 201 Archimedes’ screw, 6 surface with directrix ellipse and generatrix sinusoid, 205 Arbitrary tubular surface, 364 surface with directrix sinusoid and generatrix cycloid, 202 Astroidal ellipsoid, 681 surface with directrix sinusoid and generatrix ellipse, 204 Astroidal helicoid, 246 surface with directrix sinusoid and generatrix parabola, 204 Astroidal surface of revolution, 121 surfaces of general type, 199 Astroidal torus, 121 Cassini surfaces, 217 Asymototical cone, 115 Catalan’s surface, 435 Catalan surfaces, 1, 72, 77, 92, 213, 264, 435 Catenoid, 100, 152, 155, 428 B classical, 153 Ball’s cylindroid, 94 Cayley surface, 165 Barrel-shaped surface of revolution, 125, 139 Central surfaces, XVII, 613 Base of a cyclic surface, 342 “Chair”, 629 Basic cylinder, 249 Channel surfaces, 347, 355 Basis ruled surface, 233 Chen–Gackstatter surfaces, 449, 452 Beltrami surface, 108 Circular cylindrical surfaces, 150, 155, 357 Bi-agnesienne translation surface, 175 Circular helical surface with Bianchi surfaces, XVII generating circles lying in the osculating plane of a helix of centers Bicatenary surface of translation, 166 of the circles, 228, 238 Bicosine surface of translation, 208 generatrix circles in the planes of pencil, 225, 234 Bicubic surface of Bézier, 327 Circular helical surfaces, 232, 233 Bicycloidal surface of translation, 173 Circular surface of translation, 164 Bi-elliptical surface of translation, 170, 174 Circular torus, 111, 113, 130, 234, 245, 301, 358 Bihyperbolic transltion sutface, 175 Closed, 00 Biquadratic surface of translation, 165 circular torus Bisemicubic surface of translation, 209 h–coherent surface, 417

S.N. Krivoshapko and V.N. Ivanov, Encyclopedia of Analytical Surfaces, 745 DOI 10.1007/978-3-319-11773-7, © Springer International Publishing Switzerland 2015 746 Surfaces

one-sided surface without singular points, 422 Cyclic surface in a cylinder, 302 quadratic surface, 625 Cyclic surface with circles in the planes of ruled helical surface, 229, 250 meridians of the sphere and with a center-to-center line on the same torus, 111, 113 sphere, 400 “Clover knot”, 402 pencil and with a straight line of centers, 408 Coaxial surfaces of revolution, 213 pencil and with a waving line of centers on a cylinder, 400 Combined bicubic surface of Bezier, 328 Cyclic surface with a generatrix circle in the planes of pencil and with a Complete Abelian minimal surfaces, 449 plane Complete embedded H–surface, 483 centerline in the form of the 2nd order curve, 407 Complete minimal surfaces, 449 line of centers in the form of a logarithmic spiral, 405 Complete orientable minimal surfaces, 452 line of centers in the form of spiral of Archimedes, 407 Conical helicoid of constant slope, 8 Cyclic surface with generatrix circles Conical surface, 59 of variable radius and with plane line of centers constructed about of revolution, 70, 75, 619 circular cylinder, 406 of the n-order, 60, 66 Cyclic surfaces, 168, 207, 339, 341 Conical surface with of revolution, 339, 412, 413 directrix Agnesi curve, 66 Cyclic surfaces with directrix curve in the form of Cartesian folium, 61, 67 circles in planes of pencil, 283, 394 directrix curve on a sphere, 61, 71 plane of parallelism, 168, 213, 385 Conic limaçons of revolution, 196, 567 Cyclides with a triple point, 651 Connecting canal for two cylindrical surfaces with parallel axes, 381 Cycloidal surface of translation, 173 Conoid, 72, 77, 81 Cylindrical evolvent, 88 helical limaçon with parabolic generatrix, 596, 597 helical, 429 helical strip, 41, 233 parabolic, 82 helical strip of variable pitch, 252 of Plücker, 89 ruled rotational limaçon, 7 right sinusoidal, 86 surface, 37, 38, 60 with a directrix catenary, 85 surface of general type, 38 with a directrix circle, 81, 83 surface of revolution, 618 with two axes, 616 Cylindrical-and-conical spiral strip, 53 Constant section twist compensated generalized cylinder, 185 Cylindrical-and-spherical spiral-shaped strip, 77, 284 Continuous topographic ruled surface with distributing ellipse, 91 Cylinder Continuous topographic surface of Cassini, 217 epicycloidal, 575 Continuous topographic surfaces, 213 hypocycloidal, 576 of different series, 213 oblique, 38 of one series, 213 right, 38 Contour surface, 519 Cylindroid, 72, 77, 80, 92 Convex polyhedron, 677 with parabola and sinusoid lying on parallel ends, 95 Convolute helicoid, 78, 230, 568 with two directrix circles lying in mutually perpendicular planes, Coons surfaces, 547 93, 94 Coons surfaces on a curvilinear quadrangular plane, 522, 548 with two directrix ellipses, 93 Costa minimal surface, 437 “Cornucopia”, 283 Corrugated D cylinder, 136 “Darbazi surface”, 182 paraboloid of revolution, 507, 508 Darboux surface, XVII sphere, 508, 509 Deformed sphere, 122 surface of revolution of a general sinusoid, 117, 135, 206, 243 Delaunay surface, 155 surfaces, 485 “Dervish”, 630, 656 Cross cap, 419, 423 Developable surfaces, 1, 2 Crossed trough, 529 of the third and fourth order, 2, 31 “C”–smooth (continuous) surface, 328 with two plane directrix curves, 20 Cubic Cayley surface, 165 Diagonal cubic surface of Fermat, 633, 665 Cubic cone, 634 Diagonal cosine surface of translation, 209 Cubic surface x3 +xy+z=0, 632 Diagonal quartic surfaces, 175 Cubic surface with 24 straight lines, 635 Diagonal surface of translation, 175 Cuboid, 454 Dini’s helicoid, 239 Cyclic helical surface with a line of centers of variable pitch, 254 Ding–Dong surface, 141 Cyclic limaçons of revolution, 567 Directrix surface, 185 Cyclic cylindrical helical limaçon, 567, 592, 593 Double Blutel conic surface, 315 Cyclic minimal surface, 339 Double sphere, 646 Cyclic rotational surface with axoids “cylinder–plane”, 571 “Drop”, 143 Surfaces 747

Drop-shaped reservoir, 148 Globoidal worn gearing, 130 Dupin cyclides of the Goursat’s surface, XVII, 642, 644, 678 ﬁrst type (of the forth order), 353 the G surface, 438 second type (of the third order), 352, 355 Gyroid, 438, 447 third type but of the forth order, 353

H E “Handkerchief surface”, 631 Edlinger’s surfaces, 545 Harmonic minimal surfaces, 555 “Eight”, 53 Harmonic surface of right translation, 555 “Eight surface”, 141 Harmonic surface of right translation of a sinusoid with changing Elementary amplitude, 553 rotational bicubic surface of Bezier, 328 Hashimoto surfaces, 480 ruled surface, 1 Heart surface, 660 surface of Bezier, 327 Helical Ellipsoid, 622 cone, 72 of revolution, 140, 322, 622 cylindroid, 92, 430 Elliptic limaçon, 592 helicoid, 296 pole, 237 paraboloid, 163, 176, 459, 623 sinusoidal strip, 243, 245 surface of translation, 170 torse, 6 torus, 112 torus, 234 Elliptical twisted surface with circles in the planes of pencil, 313 conical surface, 62 Helical surface, 207, 225 cylinder, 94 formed by the tangents of helix, 701 velaroid, 183, 648 generated by binormals of a cylindrical helix, 231 Embedded triply-periodic minimal surfaces, 455 with generatrix curve in the form of evolvent of the circle, 246 Enneper’s surface, 432, 459 with generatrix curve in the form of hyperbola, 246, 248 Epicycloidal cylinder, 575 with generatrix cycloid, 247 Epitrochoidal surface, 347 with generatrix ellipse, 244, 313 Equidistance of the system with parabolic generatrix of arbitrary position, 240 “point–cylinder”, 687 with sinusoidal generatrix, 242 “straight–cylinder”, 685 Helical surfaces “straight–sphere”, 686 of variable helical parameter, 568 “straight–torus”, 688 of variable pitch, 225, 249 Escher surface, 358 Helicoid Euler special cones, 691 developable conic, 8 Euler surface of the forth order, 636 developable with a cuspidal edge on the paraboloid of revolution, 9 Evolute surface of carved surface, 186 evolvent, 230 Evolvent conical surface, 61, 67 open evolvent, 5 Evolvent cylindrical surface, 48 pseudo-developable, 79, 210, 230 Evolvent surface, XVII right convolute, 230 spherical, 76 Helix-shaped F preliminarily twisted strip, 298 Fairing of cycloidal type, 107 preliminarily twisted surface of circular cross section, 309 Falling Drop, 129 preliminarily twisted surface of elliptical cross section, 308, 309 Flat saddle in the drum, 564 surface with generatrix circle of variable radius lying on a plane, Fractal surfaces, 692 300 Frezier’s cylindroid, 96 surface with variable elliptical cross section, 299 Funicular surface, 160, 179 surfaces, 293, 372 “Funnel”, 553 twisted strip with straight generatrixes in the planes of pencil, 296 twisted surface of elliptical cross section in the planes of pencil, 312 Henneberg minimal surface, 439, 446 G Heptahedron, 417 Gabriel’s Horn, 119 Hoffman’s minimal surface, 709 Gabriel’s Trumpet, 119 Honeycomb conical surface, 495 General double curved ruled surface, 72 Horn torus, 111 General ruled surface, 72 “Hornlet”, 403 General surfaces of revolution of parabola, 123 Hunt ‘s surface, 661 General Weddle surface W, XVII Hyperbolic-and- logarithmic surface of revolution, 126 Generalized surface of translation, 160 Hyperbolic Peterson surfaces, 325 cylinder, 39, 45 Veronese surface, 317 conical surface, 59, 68 Generic helical ruled surface, 229 helicoid, 282 Globoid, 130 octahedron, 681 748 Surfaces

paraboloid, 1, 72, 81, 163, 178, 320, 555, 615, 616 meridian in the form of a cycloid, 194 Hyperboloid of two sheets, 624 parabolic meridian, 190 Hypocycloidal cylinder, 576 with a sinusoid as meridian, 191 Monge ruled surface with the circular cylindrical directing surface, 6, 214 I Monkey saddle, 564 Icosahedron, 677 Morse cone, 620 Imaginary surface of the second order, 613, 627 “Isolator”, 136 Isometric surfaces, 543 N Narrowing saddle surface of Rosendorn, 563 Neovius’ surface, 434 J Nodoid, 152, 155, 156 Jet Surface, 100 Nodoid surface connecting two circular cones, 157 Joachimsthal canal surfaces, 344 Non-central surface, 613 Joachimsthal cosine canal surfaces of the Non-orientable surface, 418 1st type, 348 Non-orientable one-sided surface, 423 2nd type, 349 Nonholonomic rotation surface, 153 3rd type, 351 Nordstrand’s weird surface, 638 Normal cyclic helix-shaped surface consisting of identical elements, 377 K Normal cyclic surface with Kappa surface, 217 plane circular line of centers and with a generator circle of variable Kinematical surfaces of general type, 196, 362, 567 radius, 376 Kinematic ruled surface on the base of one-sheet hyperboloidal surfaces Normal cyclic surfaces, 356, 357, 365 of revolution Normalie, 343, 355 as ﬁxed and loose axoids (one axoid is located outside another), 610 as ﬁxed and loose axoids (one axoid is located in the interior of another), 611 O “Kiss surface”, 141 Oblate Klein bottle, 419, 422 ellipsoid of revoluton, 139, 150 Klein surface, 422 spheroid, 139 The K–surfaces, 474 Oblique Kuen surface, 481 circular conical surface, 59, 64 Kuen surfaces, 691 circular cylinder, 54, 55 Kummer surface, 645 cone, 59 ellipsoid of revolution, 113 elliptical conical surface, 65 L elliptical cylinder, 56 Lemniscate cylinder, 48 helicoid, 75, 568 Lemniscate one-sided surface, 425 hyperbolic cylinder, 58 “Lemon”, 112 hyperbolic paraboloid, 615 Lichtenfels minimal surface, 442 parabolic cylinder, 11, 57 Lidinoid, 438, 448 parabolic surface of translation, 178 Liouville surface, XVII ruled surface, 77 Locally isometric surfaces, 430 Octahedron, 417, 677 Lopez Minimal Surface, 449 Oliveira’s minimal surface, 449 One-sheet elliptical paraboloids, XVIII One-sheet hyperboloid, 621 M One-sheet hyperboloid of revolution, 1, 75, 105 Menn’s surface, 638 One-sided ruled surface, 418 Middle surface of non-bending shell of revolution under uniform One-sided surfaces, 417 pressure, 150 Open elliptic torus, 113 Minding surfaces of constant negative Gaussian curvature, 479 Open right helicoid, 78 Minimal Lopez–Ros surface, 448 Open ruled helical surface, 229, 250 Minimal surface of Hoffman, 449 Open torus, 111 Minimal surface of revolution, 100, 428 Ordinary helical surfaces, 225, 227, 239 Minimal surfaces, 427 Ortoid ruled surfaces, 691 of Peterson, 450 Overlimited parabolic velaroid, 641 of Thomsen, 451, 459 with free boundaries, 427, 445 “Miter surface”, 627 P Möbius strip, 418, 422 “Paper Bag”, 216 Monge surface with cylindrical directrix surface and Parabolic-and- logarithmic surface of revolution, 125 hyperbolic meridian, 193 Parabolic meridian in the form of catenary, 195 conic surface, 617 Surfaces 749

cylinder, 39, 43 Quasi-helical cyclic surface with boundary circles given in advance, humming-top, 120 303 rotational surface with axoids “plane-cylinder”, 511, 512 Quassi-polyhedrons, 678 spiroidal surface with axoids “plane-cylinder”, 605 surface of conoidal type, 658 torus, 125 R torse, 10 Rectangular parallelepiped, 332, 336 velaroid, 182 Regular Paraboloid of revolution, 9, 10, 109, 163, 174 cyclic cylindrical rotational surface, 599 of the fourth order, 127, 128 cyclic cylindrical spiroidal surface, 595 with radial waves, 524 cyclic rotational surface, 714 Peano saddle, 562, 563 ruled cylindrical spiroidal surface, 594 Pedal surface, 650, 655 Rembs’ surface, 476 “Pear”, 133 Richmond’s minimal surface, 443 Peninsula surface, 657 Riemann-Schwartz surface, 427, 433 “PenKa”, 152, 154 Riffled surfaces, 485 Perseus paraboloid, 412 Riffle-shaped surfaces, 485 Peterson surfaces, 325, 343 Right astroidal cylindrical surface, 47 Pillow Shape, 648 Right circular Piriform Surface, 142 conical surface, 60 Planar Enneper, 447 helical surface, 211, 232, 236, 390 Plane, XIII spiral-shaped surface with a generatrix circle of variable Plücker conoid, 89, 91, 95, 691 radius, 391 Polar surface of a given curve, XVII spiral surface, 386 Polyhedrons, 677 surface on a cylinder, 392 Pontryagin’s surfaces, 691 Right conical surface with a plane director curve in the form of circular Poweroid, 127 sinusoid, 69 Preliminarily twisted circular torus, 397 Right conical sinusoidal wave-shaped surface, 492 Prismatoid, 677 Right conoid, 78, 81 Prolate ellipsoid of revolution, 139 with directrix parabola the axis of which is parallel to axis of Prolate spheroid, 139 conoid, 78, 87 Pseudo-catenoid, 132 Right convolute helicoid, 230 Pseudo-developable helicoid, 78, 210, 230 Right cylindrical surface with directrix cycloid, 510 Pseudo-developable helix-shaped surface with variable pitch, 306 Right cylindrical surface with directrix logarithmic spiral, 49 Pseudo-sphere, 108, 239 Right cylindrical surface with directrix semi-cubical parabola, 46 Punctured sphere with a handle, 449 Right cylinder with directrix Pyramid, 60, 677 catenary, 46 hyperbolic spiral, 49 logarithmic spiral, 49, 53 Q spiral of Archimedes, 51 Quadratic surfaces, 115, 613, 614, 625 spiral of Fermat, 50 Quadrics, 315, 613, 625 Right elliptical cylindrical surface, 56 with a double conic, 651 Right helical conoid, 429 with a double straight line, 655 Right helicoid, XVIII, XVIII72, 81, 210, 296, 297, 429 with three double straight lines, 652 Right helicoidal surface with variable pitch, 253 “Quail Egg”, 142 Right hyperboloid, 105 Quartic non-orientable surface, 421 Right-side right helicoid, 429 Quartic surface with Right wave-shaped cylindrical surface, 497 double conic and with a double straight line, 650 Right waving helicoid, 500 double straight line and a triple point, 655 Roman surface, 421, 423 parabola, ellipse, parabola in three principal coordinate sections, Rotational oblique helicoid, 307 647 Rotational surfaces, 207, 568 4th order curve, ellipse, the 4th order curve in three principal coor- with axoids “cone-cone”, 580 dinate sections, 647 with axoids “cone-plane”, 592, 714 triple straight line, 650 with axoids “cylinder-cylinder”, 573 two double straight lines, 651 with axoids “cylinder-plane”, 569 Quartoid, 127 with axoids “plane-cone”, 307, 587 Quasi-ellipsoidal surface with with axoids “plane-cylinder”, 510-512, 603-605 concave segments between ribs, 333 Rotational surface with axoids “cone-cone” formed by convex segments and a cylindrical insertion, 336 parabola in the process of external rolling (the first type), 586 convex segments between ribs, 335 parabola in the process of external rolling (the second type), 586 three values of semi-axes, 331 straight intersecting the axis of a mobile cone in the process of Quasi-ellipsoidal surfaces, 331 external rolling, 584 with cylindrical insertions, 336 straight parallel to the axis of a mobile cone in the process of with six values of semi-axes, 333 external rolling, 585 750 Surfaces

Rotational surface with axoids “cone-cone” generated by a straight line Spherical one-sheet saddle surfaces, 561 coming through the common vertex of the axoids Spherical surface, 475 external rolling, 581 of the third order, XVII internal rolling, 583 Spheroid, 139 Rotational surface with axoids “cylinder-cylinder” formed by a straight Spindle-shaped cyclide, 649 intersecting the axis of a mobile cylinder in the process of external Spiral conical strip, 61, 70, 266 rolling, 579 Spiral screw, 289 intersecting the axis of a mobile cylinder in the process of internal Spiral-shaped surface, 275 rolling, 579 “Shell without vertex”, 290 not intersecting the axis of a mobile cylinder in the process of “Shell with vertex, 290 external rolling, 577 with generatrix sinusoids and with a directrix line of constant pitch not intersecting the axis of a mobile cylinder in the process of on a circular cone, 503 internal rolling, 578 Spiral-shaped surfaces, 259, 275 Rotational surface with axoids “plane-cone”, 587 Spiral surface with formed by a straight intersecting the axis of a mobile cone, 590 directrix logarithmic spiral and with parabolic generatrix, 273 formed by a straight parallel to the axis of a mobile cone, 591 generatrix ellipse, 267 generated by a straight line coming through the vertex of a mobile generatrix in the form of cycloid, 270 cone, 588 generatrix in the form of evolvent of a circle, 272 Ruled conic limaçon of revolution, 197 hyperbolic generatrix, 269 Ruled cylindrical helical limaçon, 593 parabolic generatrix of arbitrary position, 273 Ruled helical surfaces, 225, 229 sinusoidal generatrix, 270 Ruled improper affine sphere, 166 straight generatrixes in the planes of pencil, 76, 265, 272 Ruled minimal surface, 72, 77 Spiral surfaces, 259, 261 Ruled quasi-ellipsoidal surface, 334 Spiroidal ruled surfaces of Rachkovskaya-Kharabaev, 606 Ruled rotational surface with axoids “developable conical helicoid-right circular cone”, 608 of Lusta, 574, 652 with axoids “evolvent helicoid-right circular cone”, 607 with axoids “cylinder-plane”, 570 with axoids “evolvent helicoid-right elliptical cone”, 609 with axoids “plane-cylinder”, 510 Spiroidal surfaces, 592 Ruled rotor cylindroid, 80 with axoids “cylinder-cylinder”, 597 Ruled saddle surfaces, 72 with axoids “cylinder-plane”, 593 Ruled surfaces, 1 with axoids “plane-cone”, 602 of the 4th order with three double lines, 652 with axoids “plane-cylinder”, 603 of negative Gaussian curvature, 1, 72, 264, 296, 306 Spiroidal surfaces with axoids “cylinder-cylinder” formed by a straight of revolution of negative Gaussian curvature, 1, 72, 105 intersecting the axis of a mobile cylinder in the process of external of zero Gaussian curvature, 2 rolling, 600 intersecting the axis of a mobile cylinder in the process of internal rolling, 601 S not intersecting the axis of a mobile cylinder in the process of Saddle in the drum, 81 external rolling, 599 Saddle surface of the ε class, 561 not intersecting the axis of a mobile cylinder in the process of Saddle surfaces, 561 internal rolling, 599 Saddle surface with zero rotation of horn, 565 Spiroidal surface with axoids “plane-cone» formed by a straight passing “Saint Elias surface”, 399 through the vertex of a mobile cone, 602 “Seashell”, 288 Spiroidal surface with axoids “plane-cylinder» formed by a straight Scherk’s minimal surface (the first one), 431 intersecting the axis of a rolling cylinder, 605 Scherk’s minimal surface, the second, 443 parallel to the axis of a rolling cylinder, 604 Schwarz surface, 433 Środa’s parabolic surface, 639 the Schwarz P and D surfaces, 438 Standard elementary surface of Bezier, 328 Self-intersecting surface, 78, 432, 478, 691 Steinbach screw, 280 Shoe surface, 171 Steiner ellipsoid, 691 Shtaerman’s surface with plane contour, 182 Steiner surfaces of the first and second types, 654 Sievert’s surface, 478 Supercyclides, 315, 353 Sine surface, 659 Super-ellipsoid, 332 Sinusoidal Super-quadrics, 332 cylindrical surface, 136, 503 Super-toroid, 332 helicoid, 491 Surface formed by velaroid, 179, 491 rotation of a meridian in the form of semi-cubical parabola, 118 Skew ruled surface, 72 cubic parabolas but with radial waves damping in the central point, “Ski hill”, 665 527 Skidan’s ruled surface, 498 Surface Snail surface, 280 of Bezier, 327 Soucoupoid, 129 of binormals, 2, 233 Sphere, 99, 475 of Bonnet, XVII with cycloidal crimps, 513 of catenoidal type, 152, 153 with one cross-cap, 419 of circles of Feuerbach, 637 with two cross-cap, 419 of conjugation of coaxial cylinder and cone, 117 Surfaces 751

of conjugation of two coaxial cylinders of different diameters, 115 of revolution with extreme properties, 152 of cylindrical flexible hopper-type bin for the keeping of dry of right translation, 159, 163, 297, 624 materials, 52 of oblique wedge, 1, 73 of double oblique conoid, 1, 73 of the second order, 613 of double oblique cylinder, 73 of umbrella type, 519, 521 of double oblique cylindroid, 1, 97 of umbrella type with singular central point, 532 of helical pole, 390 of umbrella type with central plane point, 527 of normal of a curve, xvii with the constant equiaffine invariants, 691 of oblique cylinder, 1, 73 with one family of plane lines of principle curvatures, 471 of principal normals, 1, 2 with spherical director curve, 463 of Riemann-Schwartz, 427, 433 Synclastic surface, 474 of spiral spring, 398 of tangent lines, 2, 3 Surface of constant slope, 11, 33, 34 T with directrix circle, 33 Tanglecube, 644 with edge of regression on a paraboloid of revolution, 694 Tangent torse, 3 Surface of revolution conical surface of revolution,59, 70, 75, 101 “Egg” of the fourth order, 142 cylindrical surface of revolution, 39, 103 “Egg” of the third order, 142 Tetrahedron, 421, 677 given by the harmonic function z = ln[x2 + y2]1/2, 552 Three-axial ellipsoid of Krasovskiy, 622 of Agnesi curl, 122 Three-coherent surface, 422 of Agnesi curl (the 2nd variant), 662 Togliatti surfaces, 656 of astroid, 120 Tooth surface, 644 of biquadrate parabola, 138 Topographic surface, 213 of curve z = bexp(-a2x2) around the z axis, 114 with the given elliptical cross sections, 214, 216 of general sinusoid, 134, 243, 272 Toroid, 130 of hyperbola z = b/x around the Oz axis, 119, 322 Torse of constant slope with of parabola, 120, 123 directrix catenary, 35 of parabola of arbitrary position, 123, 137, 241 directrix curve in the form of evolvent of the circle, 694 of usual cycloid, 131 directrix ellipse, 36 with damping circular waves, 128 directrix parabola, 34 with geometrically optimal rise, 149 edge of regression on one sheet hyperboloid of revolution, 36 Surface of translation of Torse with catenary along catenary, 166 generating straight lines lying in the normal planes of a spherical circle along elliptical centerline, 389 curve, 19 circle along parabola, 167 given line of curvature in the form of the second order parabola, 17 circle along sinusoid, 387 parabola and circle in parallel planes, 27 parabola along hyperbola, 171 parabola and ellipse in parallel planes, 29 sinusoid along parabola, 169 hyperbola and parabola in parallel planes, 30 sinusoid along sinusoid, 172 two ellipses lying in parallel planes and with parallel axes, 23 Surface of velaroidal type on annulus plan with two families of sinu- two ellipses placed in mutually perpendicular planes, 31 soids, 180 two parabolas having one common axis and lying in intersecting Surface of umbrella type 519, 521 planes, 25 formed by cubic parabolas on a cycloidal plan, 531 two parabolas of the second and forth order placed in parallel planes formed by semi-cubic parabolas on a cycloidal plan, 533 and with parallel axes, 26 with a sinusoidal generatrix, 169, 191, 228, 242, 262, 489, 520 two parabolas with intersecting axes, 21 Surface with a spherical director curve and with two parabolas lying in intersecting planes but with parallel axes, 22 elliptical generatrix in the planes of meridians of the sphere, 466 Torse with an edge of regression given as parabolic generatrix in the planes of meridians of the sphere, 465 x=e–t cost; y=e–t sint, z=e–t, 13 Surfaces given by harmonic functions, 551 x = v-v3/3;y=v2; z=a(v + v3/3), 14 Surfaces Torse with an edge of regression in the form of of Archimedes, 229 hyperbolic helical line, 16 of Bézier, 327 line of intersection of circular cylinder and circular cone, 32 of Bonnet, XVII line of intersection of two cylinders with the perpendicular axes, 15 of Delaunay, 155 Torse with an edge of regression on of congruent cross-sections, 196, 207, 425, 485 ellipsoid of revolution, 11 of constant Gaussian curvature, 471 one sheet hyperboloid of revolution, 12 of constant mean curvature, 471, 483 Torse surfaces, 1-3, 198 of constant negative Gaussian curvature, 240, 479 Torses, 1, 3 of constant positive Gaussian curvature, 474 Tractrisoid, 725 of constant zero Gaussian curvature, 2 Transcendental affine minimal surface, 451, 460 of Darboux, XVII Transcendental surface with two families of plane isotropic coordinate of double oblique conoid, 1, 73 lines, 286 of Joachimsthal, 344 Translation surfaces, 159-161, 163, 431 of oblique translation, 1, 159, 175 with congruent generatrix and directrix curves, 159, 174 of plane-and-parallel translation, 207 Translation Weingarten surfaces, 160 752 Surfaces

“Trash Can”, 216 V Trinoid, 441, 446 Velaroidal surface, 162, 179, 491 Truncated Veronese surfaces, 317, 421 cone, 59, 149, 620 Vinded column, 233 ellipsoid of revolution, 150 Virich cyclic surface, 558 paraboloid of revolution, 150 Volkov’s diagonal circular surface of translation, 176 sphere, 149 Voss surface, XVIII Tubular helical surface, 232, 235, 378 Tubular helix-shaped surface with a center-to-center line of variable lead, 372 W Tubular loxodrome, 467 Wallis’s conical edge, 89, 652 Tubular spiral surface, 361 Wave-shaped surface, 485 Tubular surface on the sphere, 370 Wave-shaped torus, 400, 488 Tubular surface winding the sphere, 468 Wave-shaped torus on the sphere, 501 Tubular surface with Wave-shaped, waving, and corrugated surfaces, 169, 400, 485 line of centers on one-sheet hyperboloid of revolution, 277 Wave surface, 485 plane elliptical line of centers, 368, 384 Waving chain with elliptical cross sections limited by elliptical plane hyperbolic centerline, 366 cone, 517 plane line of centers in the form of a cycloid, 364 cylinder, 516 plane line of centers in the form of the evolvent of a circle, 362 paraboloid, 517 plane parabolic line of centers, 365 Waving conical surface in lines of principle curvatures with inner plane sinusoidal centerline, 369 vertex, 494 Tubular surfaces, 357, 359, 360 Waving ellipsoidal surface, 523 Twice canal surfaces, 352 Waving sinusoidal velaroid, 491 Twice oblique trochoid cylindroid, 97 Waving surfaces, 485 Twice oblique cylindroid, 97 Waving surface with Twice ruled surfaces, 72, 102, 105 cubical parabolas, 514 Twisted surface with congruent ellipses in parallel planes, 210 semi-cubical parabolas, 515 Twisted surface with congruent sinusoids at the parallel planes, 491 Waving surface with the pseudo Agnesi curls Two-coherent surface, 417 of cylindrical type, 504 Two-sheeted elliptical paraboloids, XVIII on a circular plan, 506 Two-sheeted hyperboloid of revolution, 114 Wavy catenoid, 447 Tzitzéica surface of the “Wedding ring”, 414 second order with the centroaffine invariant I =-a2, 320 Wedge-shaped surfaces, 81 second order with the centroaffine invariant I = a2, 321 Weingarten surfaces, 471 third order with the centroaffine invariant I = 1/27, 320 “Wellenkugel”, 116 third order with the centroaffine invariant I = -4/27, 322 Whitney umbrella, 78, 88 Tzitzéica’s surface with centroaffine invariant equal to I = -4/(27b2), the W surface, 471 119 Tzitzéica surfaces, 319 Z Zindler’s conoid, 90 U Umbrella surface formed by biquadratic parabolas and with astroidal level line, 530 Unduloid, 152, 155, 156 Usual helical surface, 592