DOCSLIB.ORG

Analytic Geometry of Space

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Analytic Geometry of Space Hmerican flDatbematical Series E. J. TOWNSEND GENERAL EDITOR MATHEMATICAL SERIES While this series has been planned to meet the needs of the student is who preparing for engineering work, it is hoped that it will serve equally well the purposes of those schools where mathe matics is taken as an element in a liberal education. In order that the applications introduced may be of such character as to interest the general student and to train the prospective engineer in the kind of work which he is most likely to meet, it has been the policy of the editors to select, as joint authors of each text, a mathemati cian and a trained engineer or physicist. The following texts are ready: I. Calculus. By E. J. TOWNSEND, Professor of Mathematics, and G. A. GOOD- ENOUGH, Professor of Thermodynamics, University of Illinois. $2.50. II. Essentials of Calculus. By E. J. TOWNSEND and G. A. GOODENOUGH. $2.00. III. College Algebra. By H. L. RIETZ, Assistant Professor of Mathematics, and DR. A. R. CRATHORNE, Associate in Mathematics in the University of Illinois. $1.40. IV. Plane Trigonometry, with Trigonometric and Logarithmic Tables. By A. G. HALL, Professor of Mathematics in the University of Michigan, and F. H. FRINK, Professor of Railway Engineering in the University of Oregon. $1.25. V. Plane and Spherical Trigonometry. (Without Tables) By A. G. HALL and F. H. FRINK. $1.00. VI. Trigonometric and Logarithmic Tables. By A. G. HALL and F. H. FRINK. 75 cents. The following are in preparation: Plane Analytical Geometry. ByL. W. DOWLING, Assistant Professor of Mathematics, and F. E. TLTRNEAURE, Dean of the College of Engineering in the University of Wisconsin. Analytic Geometry of Space. By VIRGIL SNYDER, Professor in Cornell University, and C. H. SISAM, Assistant Professor in the University of Illinois. Young and Schwartz s Elementary Geometry. By J. W. YOUNG, Professor of Mathematics in Dartmouth Col lege, and A. J. SCHWARTZ, William McKinley High School, St. Louis. HENRY HOLT AND COMPANY NEW YORK CHICAGO ANALYTIC GEOMETRY OF SPACE BY VIRGIL SNYDER, Pn.D. (GOTTINGEN) PROFESSOR OF MATHEMATICS AT CORNELL UNIVERSITY AND C. H. SISAM, PH.D. (CORNELL) ASSISTANT PROFESSOR OF MATHEMATICS AT THE UNIVERSITY OF ILLINOIS NEW YORK HENRY HOLT AND COMPANY 1914 COPYRIGHT, 1914, BY HENRY HOLT AND COMPANY J. S. Gushing Co. Berwick & Smith Co. Norwood, Mass., U.S.A. PREFACE IN this book, which is planned for an introductory course, the first eight chapters include the subjects usually treated in rectangular coordinates. They presuppose as much knowledge of algebra, geometry, and trigonometry as is contained in the major requirement of the College Entrance Examination Board, and as mu.ch plane analytic geometry as is contained in the better elementary textbooks. In this portion, proofs of theorems from more advanced subjects in algebra are supplied as needed. Among the features of this part are the development of linear systems of planes, plane coordinates, the concept of infinity, the treatment of imaginaries, and the distinction between centers and vertices of quadric surfaces. The study of this portion can be regarded as a first course, not demanding more than thirty or forty lessons. In Chapter IX tetrahedral coordinates are introduced by means of linear transformations, under which various invariant proper ties are established. These coordinates are used throughout the next three chapters. The notation is so chosen that no ambigu ity can arise between tetrahedral and rectangular systems. The selection of subject matter is such as to be of greatest service for further study of algebraic geometry. In Chapter XIII a more advanced knowledge of plane analytic geometry is presupposed, but the part involving Plticker s num bers may be omitted without disturbing the continuity of the subject. In the last chapter extensive use is made of the cal culus, including the use of partial differentiation and of the element of arc. The second part will require about fifty lessons. CONTENTS CHAPTER I COORDINATES PAGE ARTICLE 1. Coordinates 3 2. Orthogonal projection 5 3. Direction cosines of a line ........ 6 4. Distance between two points 5. Angle between two directed lines 6. Point dividing a segment in a given ratio . 7. Polar coordinates .10 8. Cylindrical coordinates n 9. Spherical coordinates CHAPTER II PLANES AND LINES 12 10. Equation of a plane 11. Plane through three points .13 . 13 12. Intercept form of the equation of a plane . ... 14 13. Normal form of the equation of a plane . \ 15 14. Reduction of a linear equation to the normal form .... 15. Angle between two planes 17 16. Distance to a point from a plane 19 17. Equations of a line . 19 18. Direction cosines of the line of intersection of two planes 19. Forms of the equations of a line . " . .20 20. Parametric equations of a line . .21 22 21. Angle which a line makes with a plane 22. Distance from a point to a line . .23 24 23. Distance between two non-intersecting lines . .... 24. System of planes through a line . .25 25. Application to descriptive geometry . : . .28 v Vi CONTENTS 26. Bundles of planes 29 27. Plane coordinates 31 28. Equation of a point . 32 29. Homogeneous coordinate of the point and of the plane ... 33 30. Equation of the plane and of the point in homogeneous coordinates 34 31. of Equation the origin. Coordinates of planes through the origin . 34 32. Plane at infinity 35 33. Lines at infinity .35 34. Coordinate tetrahedron ......... 35 35. System of four planes 36 CHAPTER III TRANSFORMATION OF COORDINATES 36. Translation 38 37. Rotation 38 38. Rotation and reflection of axes 41 39. Euler s formulas for rotation of axes 42 40. Degree unchanged by transformation of coordinates ... 42 CHAPTER IV TYPES OF SURFACES 41. Imaginary points, lines, and planes . .44 42. Loci of . equations .... 46 43. Cylindrical surfaces ......... 47 44. Projecting cylinders 47 45. Plane sections of surfaces 48 46. Cones 49 47. Surfaces of revolution 60 V CHAPTER V THE SPHERE 48. The equation of the sphere ........ 52 49. The absolute 52 50. Tangent plane 56 51. Angle between two spheres 65 52. Spheres satisfying given conditions ...... 56 53. Linear systems of spheres ... .57 54. Stereographic projection . 69 CONTENTS Vii CHAPTER VI FORMS OF QUADRIC SURFACES ARTICLE . 55. Definition of a quadric . ? . , ... Vlll CONTENTS CHAPTER IX TETRAHEDRAL COORDINATES ARTICLE 88. Definition of tetrahedral coordinates 109 89. Unit point ..... no 90. Equation of a plane. Plane coordinates .... Ill 91. Equation of a point H2 92. Equations of a line .... 112 93. Duality 113 94. Parametric of a and of . equations plane a point . .114 95. Parametric of equations a line. Range of points. Pencils of planes 115 96. Transformation of point coordinates .... 117 97. Transformation of plane coordinates .... 119 98. Projective transformations 120 99. Invariant points .... 121 100. Cross-ratio 12 i CHAPTER X QUADRIC SURFACES IN TETRAHEDRAL COORDINATES 101. Form of equation ...... 124 102. Tangent lines and planes 124 103. Condition that the tangent plane is indeterminate .... 125 104. The invariance of the discriminant . 126 105. Lines on the quadric surface 129 106. Equation of a quadric in plane coordinates 130 107. Polar planes 132 108. Harmonic property of conjugate points .... 132 109. Locus of which lie points on their own . polar planes . 133 110. Tangent cone ........ 133 111. Conjugate lines as to a quadric 134 112. Self-polar tetrahedron 135 113. referred to a self Equation -polar tetrahedron . .135 114. Law of inertia 13g 115. Rectilinear generators. Reguli 137 116. Hyperbolic coordinates. Parametric equations .... 138 117. Projection of a quadric upon a plane 139 118. Equations of the projection ...... 140 119. determined Quadrics three . by non-intersecting lines . .141 120. Transversals of four skew lines 143 121. The cone . quadric . 143 122. of a cone Projection quadric upon a plane . 145 CONTENTS IX CHAPTER XI LINEAR SYSTEMS OF QUADRICS PAGK AKTICLE 147 123. Pencil of quadrics 147 124. The \-discriminant 148 125. Invariant factors 15 126. The characteristic . .151 127. Pencil of quadrics having a common vertex 151 128. Classification of pencils of quadrics 151 129. Quadrics having a double plane in common . 151 130. Quadrics having a line of vertices in common 152 131. Quadrics having a vertex in common ...... 156 132. Quadrics having no vertex in common . I 63 133. Forms of pencils of quadrics I65 134. Line conjugate to a point 166 135. Equation of the pencil in plane coordinates 167 136. Bundle of quadrics of a 168 137. Representation of the quadrics of a bundle by the points plane 168 . 138. Singular quadrics of the bundle . 169 139. Intersection of the bundle by a plane 140. The vertex locus J / .170 171 141. Polar theory in the bundle 173 142. Some special bundles 17& 143. Webs of quadrics 175 144. The Jacobian surface of a web 177 145. Correspondence with the planes of space 177 146. Web with six basis points I80 147. Linear systems of rank r . .181 148. Linear systems of rank r in plane coordinates 181 149. Apolarity . / I86 150. Linear systems of npolar/Juadrics CHAPTER XII TRANSFORMATIONS OF SPACE 151. Projective metric I 88 152. Pole and polar as to the absolute 190 153. Equations of motion 191 154. Classification of protective transformations . 195 155. Standard forms of equations of projective transformations 196 156. Birational transformations 198 157. Quadratic transformations . .201 158. Quadratic inversion 201 159. Transformation by reciprocal radii ... 203 . -:..-.. 160. Cyclides . ... X CONTENTS CHAPTER XIII CURVES AND SURFACES IN TETRAHEDRAL COORDINATES I. ALGEBRAIC SURFACES ARTICLE PAGE 161. Number of constants in the equation of a surface .... 206 162. Notation 207 163. Intersection of a line and a surface 207 164.
Load more

Recommended publications
  • Chapter 11. Three Dimensional Analytic Geometry and Vectors
    Chapter 11. Three dimensional analytic geometry and vectors. Section 11.5 Quadric surfaces. Curves in R2 : x2 y2 ellipse + =1 a2 b2 x2 y2 hyperbola − =1 a2 b2 parabola y = ax2 or x = by2 A quadric surface is the graph of a second degree equation in three variables. The most general such equation is Ax2 + By2 + Cz2 + Dxy + Exz + F yz + Gx + Hy + Iz + J =0, where A, B, C, ..., J are constants. By translation and rotation the equation can be brought into one of two standard forms Ax2 + By2 + Cz2 + J =0 or Ax2 + By2 + Iz =0 In order to sketch the graph of a quadric surface, it is useful to determine the curves of intersection of the surface with planes parallel to the coordinate planes. These curves are called traces of the surface. Ellipsoids The quadric surface with equation x2 y2 z2 + + =1 a2 b2 c2 is called an ellipsoid because all of its traces are ellipses. 2 1 x y 3 2 1 z ±1 ±2 ±3 ±1 ±2 The six intercepts of the ellipsoid are (±a, 0, 0), (0, ±b, 0), and (0, 0, ±c) and the ellipsoid lies in the box |x| ≤ a, |y| ≤ b, |z| ≤ c Since the ellipsoid involves only even powers of x, y, and z, the ellipsoid is symmetric with respect to each coordinate plane. Example 1. Find the traces of the surface 4x2 +9y2 + 36z2 = 36 1 in the planes x = k, y = k, and z = k. Identify the surface and sketch it. Hyperboloids Hyperboloid of one sheet. The quadric surface with equations x2 y2 z2 1.
    [Show full text]
  • Pencils of Quadrics
    Europ. J. Combinatorics (1988) 9, 255-270 Intersections in Projective Space II: Pencils of Quadrics A. A. BRUEN AND J. w. P. HIRSCHFELD A complete classification is given of pencils of quadrics in projective space of three dimensions over a finite field, where each pencil contains at least one non-singular quadric and where the base curve is not absolutely irreducible. This leads to interesting configurations in the space such as partitions by elliptic quadrics and by lines. 1. INTRODUCTION A pencil of quadrics in PG(3, q) is non-singular if it contains at least one non-singular quadric. The base of a pencil is a quartic curve and is reducible if over some extension of GF(q) it splits into more than one component: four lines, two lines and a conic, two conics, or a line and a twisted cubic. In order to contain no points or to contain some line over GF(q), the base is necessarily reducible. In the main part of this paper a classification is given of non-singular pencils ofquadrics in L = PG(3, q) with a reducible base. This leads to geometrically interesting configur­ ations obtained from elliptic and hyperbolic quadrics, such as spreads and partitions of L. The remaining part of the paper deals with the case when the base is not reducible and is therefore a rational or elliptic quartic. The number of points in the elliptic case is then governed by the Hasse formula. However, we are able to derive an interesting combi­ natorial relationship, valid also in higher dimensions, between the number of points in the base curve and the number in the base curve of a dual pencil.
    [Show full text]
  • A Brief Tour of Vector Calculus
    A BRIEF TOUR OF VECTOR CALCULUS A. HAVENS Contents 0 Prelude ii 1 Directional Derivatives, the Gradient and the Del Operator 1 1.1 Conceptual Review: Directional Derivatives and the Gradient........... 1 1.2 The Gradient as a Vector Field............................ 5 1.3 The Gradient Flow and Critical Points ....................... 10 1.4 The Del Operator and the Gradient in Other Coordinates*............ 17 1.5 Problems........................................ 21 2 Vector Fields in Low Dimensions 26 2 3 2.1 General Vector Fields in Domains of R and R . 26 2.2 Flows and Integral Curves .............................. 31 2.3 Conservative Vector Fields and Potentials...................... 32 2.4 Vector Fields from Frames*.............................. 37 2.5 Divergence, Curl, Jacobians, and the Laplacian................... 41 2.6 Parametrized Surfaces and Coordinate Vector Fields*............... 48 2.7 Tangent Vectors, Normal Vectors, and Orientations*................ 52 2.8 Problems........................................ 58 3 Line Integrals 66 3.1 Defining Scalar Line Integrals............................. 66 3.2 Line Integrals in Vector Fields ............................ 75 3.3 Work in a Force Field................................. 78 3.4 The Fundamental Theorem of Line Integrals .................... 79 3.5 Motion in Conservative Force Fields Conserves Energy .............. 81 3.6 Path Independence and Corollaries of the Fundamental Theorem......... 82 3.7 Green's Theorem.................................... 84 3.8 Problems........................................ 89 4 Surface Integrals, Flux, and Fundamental Theorems 93 4.1 Surface Integrals of Scalar Fields........................... 93 4.2 Flux........................................... 96 4.3 The Gradient, Divergence, and Curl Operators Via Limits* . 103 4.4 The Stokes-Kelvin Theorem..............................108 4.5 The Divergence Theorem ...............................112 4.6 Problems........................................114 List of Figures 117 i 11/14/19 Multivariate Calculus: Vector Calculus Havens 0.
    [Show full text]
  • Quadric Surfaces
    Quadric Surfaces SUGGESTED REFERENCE MATERIAL: As you work through the problems listed below, you should reference Chapter 11.7 of the rec- ommended textbook (or the equivalent chapter in your alternative textbook/online resource) and your lecture notes. EXPECTED SKILLS: • Be able to compute & traces of quadic surfaces; in particular, be able to recognize the resulting conic sections in the given plane. • Given an equation for a quadric surface, be able to recognize the type of surface (and, in particular, its graph). PRACTICE PROBLEMS: For problems 1-9, use traces to identify and sketch the given surface in 3-space. 1. 4x2 + y2 + z2 = 4 Ellipsoid 1 2. −x2 − y2 + z2 = 1 Hyperboloid of 2 Sheets 3. 4x2 + 9y2 − 36z2 = 36 Hyperboloid of 1 Sheet 4. z = 4x2 + y2 Elliptic Paraboloid 2 5. z2 = x2 + y2 Circular Double Cone 6. x2 + y2 − z2 = 1 Hyperboloid of 1 Sheet 7. z = 4 − x2 − y2 Circular Paraboloid 3 8. 3x2 + 4y2 + 6z2 = 12 Ellipsoid 9. −4x2 − 9y2 + 36z2 = 36 Hyperboloid of 2 Sheets 10. Identify each of the following surfaces. (a) 16x2 + 4y2 + 4z2 − 64x + 8y + 16z = 0 After completing the square, we can rewrite the equation as: 16(x − 2)2 + 4(y + 1)2 + 4(z + 2)2 = 84 This is an ellipsoid which has been shifted. Specifically, it is now centered at (2; −1; −2). (b) −4x2 + y2 + 16z2 − 8x + 10y + 32z = 0 4 After completing the square, we can rewrite the equation as: −4(x − 1)2 + (y + 5)2 + 16(z + 1)2 = 37 This is a hyperboloid of 1 sheet which has been shifted.
    [Show full text]
  • Quadric Surfaces
    Quadric Surfaces Six basic types of quadric surfaces: • ellipsoid • cone • elliptic paraboloid • hyperboloid of one sheet • hyperboloid of two sheets • hyperbolic paraboloid (A) (B) (C) (D) (E) (F) 1. For each surface, describe the traces of the surface in x = k, y = k, and z = k. Then pick the term from the list above which seems to most accurately describe the surface (we haven't learned any of these terms yet, but you should be able to make a good educated guess), and pick the correct picture of the surface. x2 y2 (a) − = z. 9 16 1 • Traces in x = k: parabolas • Traces in y = k: parabolas • Traces in z = k: hyperbolas (possibly a pair of lines) y2 k2 Solution. The trace in x = k of the surface is z = − 16 + 9 , which is a downward-opening parabola. x2 k2 The trace in y = k of the surface is z = 9 − 16 , which is an upward-opening parabola. x2 y2 The trace in z = k of the surface is 9 − 16 = k, which is a hyperbola if k 6= 0 and a pair of lines (a degenerate hyperbola) if k = 0. This surface is called a hyperbolic paraboloid , and it looks like picture (E) . It is also sometimes called a saddle. x2 y2 z2 (b) + + = 1. 4 25 9 • Traces in x = k: ellipses (possibly a point) or nothing • Traces in y = k: ellipses (possibly a point) or nothing • Traces in z = k: ellipses (possibly a point) or nothing y2 z2 k2 k2 Solution. The trace in x = k of the surface is 25 + 9 = 1− 4 .
    [Show full text]
  • Stable Rationality of Quadric and Cubic Surface Bundle Fourfolds
    STABLE RATIONALITY OF QUADRIC AND CUBIC SURFACE BUNDLE FOURFOLDS ASHER AUEL, CHRISTIAN BOHNING,¨ AND ALENA PIRUTKA Abstract. We study the stable rationality problem for quadric and cubic surface bundles over surfaces from the point of view of the degeneration method for the Chow group of 0-cycles. Our main result is that a very general hypersurface X of bidegree (2, 3) in P2 × P3 is not stably rational. Via projections onto the two factors, X → P2 is a cubic surface bundle and X → P3 is a conic bundle, and we analyze the stable rationality problem from both these points of view. Also, 2 we introduce, for any n ≥ 4, new quadric surface bundle fourfolds Xn → P with 2 discriminant curve Dn ⊂ P of degree 2n, such that Xn has nontrivial unramified Brauer group and admits a universally CH0-trivial resolution. 1. Introduction An integral variety X over a field k is stably rational if X Pm is rational, for some m. In recent years, failure of stable rationality has been× established for many classes of smooth rationally connected projective complex varieties, see, for instance [1, 3, 4, 7, 15, 16, 22, 23, 24, 25, 26, 27, 28, 29, 30, 33, 34, 36, 37]. These results were obtained by the specialization method, introduced by C. Voisin [37] and developed in [16]. In many applications, one uses this method in the following form. For simplicity, assume that k is an uncountable algebraically closed field and consider a quasi-projective integral scheme B over k and a generically smooth projective morphism B with positive dimensional fibers.
    [Show full text]
  • Chapter 3 Quadratic Curves, Quadric Surfaces
    Chapter 3 Quadratic curves, quadric surfaces In this chapter we begin our study of curved surfaces. We focus on the quadric surfaces. To do this, we also need to look at quadratic curves, such as ellipses. We discuss: Equations and parametric descriptions of the plane quadratic curves: circles, ellipses, ² hyperbolas and parabolas. Equations and parametric descriptions of quadric surfaces, the 2{dimensional ana- ² logues of quadratic curves. We also discuss aspects of matrices, since they are relevant for our discussion. 3.1 Plane quadratic curves 3.1.1 From linear to quadratic equations Lines in the plane R2 are represented by linear equations and linear parametric descriptions. Degree 2 equations also correspond to curves you undoubtedly have come across before: circles, ellipses, hyperbolas and parabolas. This section is devoted to these curves. They will reoccur when we consider quadric surfaces, a class of fascinating shapes, since the intersection of a quadric surface with a plane consists of a quadratic curve. Lines di®er from quadratic curves in various respects, one of which is that all lines look the same (only their position in the plane may di®er), but that quadratic curves may truely di®er in shape. 3.1.2 The general equation of a quadratic curve The general equation of a line in R2 is ax + by = c. When we also allow terms of degree 2 in the variables x and y, i.e., x2, xy and y2, we obtain quadratic equations like x2 + y2 = 1, a circle. ² x2 + 2x + y = 3, a parabola; probably you recognize it as such if it is rewritten in the ² form y = 3 x2 2x or y = (x + 1)2 + 4.
    [Show full text]
  • 11.5: Quadric Surfaces
    c Dr Oksana Shatalov, Spring 2013 1 11.5: Quadric surfaces REVIEW: Parabola, hyperbola and ellipse. • Parabola: y = ax2 or x = ay2. y y x x 0 0 y x2 y2 • Ellipse: + = 1: x a2 b2 0 Intercepts: (±a; 0)&(0; ±b) x2 y2 x2 y2 • Hyperbola: − = 1 or − + = 1 a2 b2 a2 b2 y y x x 0 0 Intercepts: (±a; 0) Intercepts: (0; ±b) c Dr Oksana Shatalov, Spring 2013 2 The most general second-degree equation in three variables x; y and z: 2 2 2 Ax + By + Cz + axy + bxz + cyz + d1x + d2y + d3z + E = 0; (1) where A; B; C; a; b; c; d1; d2; d3;E are constants. The graph of (1) is a quadric surface. Note if A = B = C = a = b = c = 0 then (1) is a linear equation and its graph is a plane (this is the case of degenerated quadric surface). By translations and rotations (1) can be brought into one of the two standard forms: Ax2 + By2 + Cz2 + J = 0 or Ax2 + By2 + Iz = 0: In order to sketch the graph of a surface determine the curves of intersection of the surface with planes parallel to the coordinate planes. The obtained in this way curves are called traces or cross-sections of the surface. Quadric surfaces can be classified into 5 categories: ellipsoids, hyperboloids, cones, paraboloids, quadric cylinders. (shown in the table, see Appendix.) The elements which characterize each of these categories: 1. Standard equation. 2. Traces (horizontal ( by planes z = k), yz-traces (by x = 0) and xz-traces (by y = 0).
    [Show full text]
  • Analytic and Differential Geometry Mikhail G. Katz∗
    Analytic and Differential geometry Mikhail G. Katz∗ Department of Mathematics, Bar Ilan University, Ra- mat Gan 52900 Israel Email address: [email protected] ∗Supported by the Israel Science Foundation (grants no. 620/00-10.0 and 84/03). Abstract. We start with analytic geometry and the theory of conic sections. Then we treat the classical topics in differential geometry such as the geodesic equation and Gaussian curvature. Then we prove Gauss’s theorema egregium and introduce the ab- stract viewpoint of modern differential geometry. Contents Chapter1. Analyticgeometry 9 1.1. Circle,sphere,greatcircledistance 9 1.2. Linearalgebra,indexnotation 11 1.3. Einsteinsummationconvention 12 1.4. Symmetric matrices, quadratic forms, polarisation 13 1.5.Matrixasalinearmap 13 1.6. Symmetrisationandantisymmetrisation 14 1.7. Matrixmultiplicationinindexnotation 15 1.8. Two types of indices: summation index and free index 16 1.9. Kroneckerdeltaandtheinversematrix 16 1.10. Vector product 17 1.11. Eigenvalues,symmetry 18 1.12. Euclideaninnerproduct 19 Chapter 2. Eigenvalues of symmetric matrices, conic sections 21 2.1. Findinganeigenvectorofasymmetricmatrix 21 2.2. Traceofproductofmatricesinindexnotation 23 2.3. Inner product spaces and self-adjoint endomorphisms 24 2.4. Orthogonal diagonalisation of symmetric matrices 24 2.5. Classification of conic sections: diagonalisation 26 2.6. Classification of conics: trichotomy, nondegeneracy 28 2.7. Characterisationofparabolas 30 Chapter 3. Quadric surfaces, Hessian, representation of curves 33 3.1. Summary: classificationofquadraticcurves 33 3.2. Quadric surfaces 33 3.3. Case of eigenvalues (+++) or ( ),ellipsoid 34 3.4. Determination of type of quadric−−− surface: explicit example 35 3.5. Case of eigenvalues (++ ) or (+ ),hyperboloid 36 3.6. Case rank(S) = 2; paraboloid,− hyperbolic−− paraboloid 37 3.7.
    [Show full text]
  • Calculus III -- Fall 2008
    Section 12.2: Quadric Surfaces Goals : 1. To recognize and write equations of quadric surfaces 2. To graph quadric surfaces by hand Definitions: 1. A quadric surface is the three-dimensional graph of an equation that can (through appropriate transformations, if necessary), be written in either of the following forms: Ax 2 + By 2 + Cz 2 + J = 0 or Ax 2 + By 2 + Iz = 0 . 2. The intersection of a surface with a plane is called a trace of the surface in the plane. Notes : 1. There are 6 kinds of quadric surfaces. Scroll down to get an idea of what they look like. Keep in mind that each graph shown illustrates just one of many possible orientations of the surface. 2. The traces of quadric surfaces are conic sections (i.e. a parabola, ellipse, or hyperbola). 3. The key to graphing quadric surfaces is making use of traces in planes parallel to the xy , xz , and yz planes. 4. The following pages are from the lecture notes of Professor Eitan Angel, University of Colorado. Keep scrolling down (or press the Page Down key) to advance the slide show. Calculus III { Fall 2008 Lecture{QuadricSurfaces Eitan Angel University of Colorado Monday, September 8, 2008 E. Angel (CU) Calculus III 8 Sep 1 / 11 Now we will discuss second-degree equations (called quadric surfaces). These are the three dimensional analogues of conic sections. To sketch the graph of a quadric surface (or any surface), it is useful to determine curves of intersection of the surface with planes parallel to the coordinate planes.
    [Show full text]
  • Multivariable Calculus Math 21A
    Multivariable Calculus Math 21a Harvard University Spring 2004 Oliver Knill These are some class notes distributed in a multivariable calculus course tought in Spring 2004. This was a physics flavored section. Some of the pages were developed as complements to the text and lectures in the years 2000-2004. While some of the pages are proofread pretty well over the years, others were written just the night before class. The last lecture was "calculus beyond calculus". Glued with it after that are some notes from "last hours" from previous semesters. Oliver Knill. 5/7/2004 Lecture 1: VECTORS DOT PRODUCT O. Knill, Math21a VECTOR OPERATIONS: The ad- ~u + ~v = ~v + ~u commutativity dition and scalar multiplication of ~u + (~v + w~) = (~u + ~v) + w~ additive associativity HOMEWORK: Section 10.1: 42,60: Section 10.2: 4,16 vectors satisfy "obvious" properties. ~u + ~0 = ~0 + ~u = ~0 null vector There is no need to memorize them. r (s ~v) = (r s) ~v scalar associativity ∗ ∗ ∗ ∗ VECTORS. Two points P1 = (x1; y1), Q = P2 = (x2; y2) in the plane determine a vector ~v = x2 x1; y2 y1 . We write here for multiplication (r + s)~v = ~v(r + s) distributivity in scalar h − − i ∗ It points from P1 to P2 and we can write P1 + ~v = P2. with a scalar but usually, the multi- r(~v + w~) = r~v + rw~ distributivity in vector COORDINATES. Points P in space are in one to one correspondence to vectors pointing from 0 to P . The plication sign is left out. 1 ~v = ~v the one element numbers ~vi in a vector ~v = (v1; v2) are also called components or of the vector.
    [Show full text]
  • Quadric Surfaces the University of Kansas 1 / 23 2 2 2 X  Y  Ellipse of Semi-Major, Semi-Minor Axes A, B: a + B = 1
    1 circle of radius r: x 2 + y 2 = r 2 2 2 2 x y ellipse of semi-major, semi-minor axes a; b: a + b = 1 2 2 3 x y hyperbola of semi-major, semi-minor axes a; b: a − b = 1 4 parabola: y = ax 2 Considering these conic sections as three-dimensional objects by letting z vary, we obtain the family of cylinders. Conic Sections Recall the usual conic sections from the Cartesian plane. MATH 127 (Section 12.6) Quadric Surfaces The University of Kansas 1 / 23 2 2 2 x y ellipse of semi-major, semi-minor axes a; b: a + b = 1 2 2 3 x y hyperbola of semi-major, semi-minor axes a; b: a − b = 1 4 parabola: y = ax 2 Considering these conic sections as three-dimensional objects by letting z vary, we obtain the family of cylinders. Conic Sections Recall the usual conic sections from the Cartesian plane. 1 circle of radius r: x 2 + y 2 = r 2 MATH 127 (Section 12.6) Quadric Surfaces The University of Kansas 1 / 23 2 2 3 x y hyperbola of semi-major, semi-minor axes a; b: a − b = 1 4 parabola: y = ax 2 Considering these conic sections as three-dimensional objects by letting z vary, we obtain the family of cylinders. Conic Sections Recall the usual conic sections from the Cartesian plane. 1 circle of radius r: x 2 + y 2 = r 2 2 2 2 x y ellipse of semi-major, semi-minor axes a; b: a + b = 1 MATH 127 (Section 12.6) Quadric Surfaces The University of Kansas 1 / 23 4 parabola: y = ax 2 Considering these conic sections as three-dimensional objects by letting z vary, we obtain the family of cylinders.
    [Show full text]