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