4.7. QUADRIC SURFACES (QUADRICS) 143 The distance between them can be calculated by the formula d =   [(r 1 – r 2 )R 1 R 2 ]   |R 1 × R 2 | =     x 1 – x 2 y 1 – y 2 z 1 – z 2 l 1 m 1 n 1 l 2 m 2 n 2         l 1 m 1 l 2 m 2    2 +    m 1 n 1 m 2 n 2    2 +    n 1 l 1 n 2 l 2    2 .(4.6.3.30) The condition that the determinant in the numerator in (4.6.3.30) is zero is the condition for the two lines in space to meet. Remark 1. The numerator of the fraction in (4.6.3.30) is the volume of the parallelepiped spanned by the vectors r 1 – r 2 , R 1 ,andR 2 , while the denominator of the fraction is the area of its base. Hence the fraction itself is the altitude d of this parallelepiped. Remark 2. If the lines are parallel (i.e., l 1 = l 2 = l, m 1 = m 2 = m,andn 1 = n 2 = n,orR 1 = R 2 = R), then the distance between them should be calculated by formula (4.6.3.29) with r 0 replaced by r 2 . 4.7. Quadric Surfaces (Quadrics) 4.7.1. Quadrics (Canonical Equations) 4.7.1-1. Central surfaces. A segment joining two points of a surface is called a chord. If there exists a point in space, not necessarily lying on the surface, that bisects all chords passing through it, then the surface is said to be central and the point is called the center of the surface. The equations listed below in Paragraphs 4.7.1-2 to 4.7.1-4 for central surfaces are given in canonical form; i.e., the center of a surface is at the origin, and the surface symmetry axes are the coordinate axes. Moreover, the coordinate planes are symmetry planes. 4.7.1-2. Ellipsoid. An ellipsoid is a surface defined by the equation x 2 a 2 + y 2 b 2 + z 2 c 2 = 1,(4.7.1.1) where the numbers a, b,andc are the lengths of the segments called the semiaxes of the ellipsoid (see Fig 4.52a). Y Y ZZ()a ()b X X a a a a b b c c b b c c O O Figure 4.52. Triaxial ellipsoid (a) and spheroid (b). 144 ANALYTIC GEOMETRY If a ≠ b ≠ c, then the ellipsoid is said to be triaxial,orscalene.Ifa = b ≠ c, then the ellipsoid is called a spheroid; it can be obtained by rotating the ellipse x 2 /a 2 + z 2 /c 2 = 1, y = 0 lying in the plane OXZ about the axis OZ (see Fig. 4.52b). If a = b > c, then the ellipsoid is an oblate spheroid,andifa = b < c, then the ellipsoid is a prolate spheroid.If a = b = c, then the ellipsoid is the sphere of radius a given by the equation x 2 + y 2 + z 2 = a 2 . An arbitrary plane section of an ellipsoid is an ellipse (in a special case, a circle). The volume of an ellipsoid is equal to V = 4 3 πabc. Remark. About the sphere, see also Paragraph 3.2.3-3. 4.7.1-3. Hyperboloids. A one-sheeted hyperboloid is a surface defined by the equation x 2 a 2 + y 2 b 2 – z 2 c 2 = 1,(4.7.1.2) where a and b are the real semiaxes and c is the imaginary semiaxis (see Fig. 4.53a). A two-sheeted hyperboloid is a surface defined by the equation x 2 a 2 + y 2 b 2 – z 2 c 2 =–1,(4.7.1.3) where c is the real semiaxis and a and b are the imaginary semiaxes (see Fig 4.53b). Y Y ZZ()a ()b X X a c b O O c Figure 4.53. One-sheeted (a) and two-sheeted (b) hyperboloids. A hyperboloid approaches the surface x 2 a 2 + y 2 b 2 – z 2 c 2 = 0, which is called an asymptotic cone,infinitely closely. A plane passing through the axis OZ intersects each of the hyperboloids (4.7.1.2) and (4.7.1.3) in two hyperbolas and the asymptotic cone in two straight lines, which are the 4.7. QUADRIC SURFACES (QUADRICS) 145 Y Z X a c b O Figure 4.54. A cone. asymptotes of these hyperbolas. The section of a hyperboloid by a plane parallel to OXY is an ellipse. The section of a one-sheeted hyperboloid by the plane z = 0 is an ellipse, which is called the gorge or throat ellipse. For a = b, we deal with the hyperboloid of revolution obtained by rotating a hyperbola with semiaxes a and c about its focal axis 2c (which is an imaginary axis for a one-sheeted hyperboloid and a real axis for a two-sheeted hyperboloid). If a = b = c, then the hyperboloid of revolution is said to be right, and its sections by the planes OXZ and OY Z are equilateral hyperbolas. A one-sheeted hyperboloid is a ruled surface (see Paragraph 4.7.1-6). 4.7.1-4. Cone. A cone is a surface defined by the equation x 2 a 2 + y 2 b 2 – z 2 c 2 = 0.(4.7.1.4) The cone (see Fig. 4.54) defined by (4.7.1.4) has vertex at the origin, and for its base we can take the ellipse with semiaxes a and b in the plane perpendicular to the axis OZ at the distance c from the origin. This cone is the asymptotic cone for the hyperboloids (4.7.1.2) and (4.7.1.3). For a = b, we obtain a right circular cone. A cone is a ruled surface (see Paragraph 4.7.1-6). Remark. About the cone, see also Paragraph 3.2.3-2. 4.7.1-5. Paraboloids. In contrast to the surfaces considered above, paraboloids are not central surfaces. For the equations listed below, the vertex of a paraboloid lies at the origin, the axis OZ is the symmetry axis, and the planes OXZ and OY Z are symmetry planes. An elliptic paraboloid (see Fig 4.55a) is a surface defined by the equation x 2 p + y 2 q = 2z,(4.7.1.5) 146 ANALYTIC GEOMETRY where p > 0 and q > 0 are parameters. Y Y ZZ()a ()b X X O O Figure 4.55. Elliptic (a) and hyperbolic (b) paraboloids. The sections of an elliptic paraboloid by planes parallel to the axis OZ are parabolas, and the sections by planes parallel to the plane OXY are ellipses. For example, let the parabola x 2 = 2pz, y = 0, obtained by the section of an elliptic paraboloid by the plane OXZ be fixed and used as the directrix, and let the parabola x 2 = 2qz, x = 0, obtained by the section of the elliptic paraboloid by the plane OY Z be movable and used as the generator. Then the paraboloid can be obtained by parallel translation of the movable parabola (the generator) in a given direction along the fixed parabola (the directrix). If p = q,thenwehaveaparaboloid of revolution, which is obtained by rotating the parabola 2pz = x 2 lying in the plane OXZ about its axis. The volume of the part of an elliptic paraboloid cut by the plane perpendicular to its axis at a height h is equal to V = 1 2 πabh, i.e., half the volume of the elliptic cylinder with the same base and altitude. A hyperbolic paraboloid (see Fig 4.55b) is a surface defined by the equation x 2 p – y 2 q = 2z,(4.7.1.6) where p > 0 and q > 0 are parameters. The sections of a hyperbolic paraboloid by planes parallel to the axis OZ are parabolas, and the sections by planes parallel to the plane OXY are hyperbolas. For example, let the parabola x 2 = 2pz, y = 0, obtained by the section of the hyperbolic paraboloid by the plane OXZ be fixed and used as the directrix, and let the parabola x 2 =–2qz, x = 0, obtained by the section of the hyperbolic paraboloid by the plane OY Z be movable and used as the generator. Then the paraboloid can be obtained by parallel translation of the movable parabola (the generator) in a given direction along the fixed parabola (the directrix). A hyperbolic paraboloid is a ruled surface (see Paragraph 4.7.1-6). 4.7.1-6. Rulings of ruled surfaces. A ruled surface is a surface swept out by a moving line in space. The straight lines forming a ruled surface are called rulings. Examples of ruled surfaces include the cone (see Paragraph 3.2.3-2 and 4.7.1-4), the cylinder (see Paragraph 3.2.3-1), the one-sheeted hyperboloid (see Paragraph 4.7.1-3), and the hyperbolic paraboloid (see Paragraph 4.7.1-5). 4.7. QUADRIC SURFACES (QUADRICS) 147 The cone (4.7.1.4) has one family of rulings, αx = βy,  α 2 a 2 + β 2 b 2 ab x = β c z. Properties of rulings of the cone: 1. There is a unique ruling through each point of the cone. 2. Two arbitrary distinct rulings of the cone meet at the point O(0, 0, 0). 3. Three pairwise distinct rulings of the cone are not parallel to any plane. The one-sheeted hyperboloid (4.7.1.2) has two families of rulings: α  x a + z c  = β  1 + y b  , β  x a – z c  = α  1 – y b  ; γ  x a + z c  = δ  1 – y b  , δ  x a – z c  = γ  1 + y b  . (4.7.1.7) One of these families is shown in Fig. 4.56a. Y Y ZZ ()a ()b X X O O Figure 4.56. Families of rulings for one-sheeted hyperboloid (a) and for hyperbolic paraboloid (b). Properties of rulings of the one-sheeted hyperboloid: 1. In either family, there is a unique ruling through each point of the one-sheeted hyper- boloid. 2. Any two rulings in different families lie in a single plane. 3. Any two distinct rulings in the same family are skew. 4. Three distinct rulings in the same family are not parallel to any plane. The hyperbolic paraboloid (4.7.1.6) has two families of rulings: α  x √ p + y √ q  = 2β, β  x √ p – y √ q  = αz; γ  x √ p + y √ q  = δz, δ  x √ p – y √ q  = 2γ. (4.7.1.8) One of these families is shown in Fig. 4.56b. 148 ANALYTIC GEOMETRY Properties of rulings of a hyperbolic paraboloid: 1. In either family, there is a unique ruling through each point of the hyperbolic paraboloid. 2. Any two rulings in different families lie in a single plane and meet. 3. Any two distinct rulings in the same family are skew. 4. All rulings in either family are parallel to a single plane. 4.7.2. Quadrics (General Theory) 4.7.2-1. General equation of quadric. A quadric is a set of points in three-dimensional space whose coordinates in the rectangular Cartesian coordinate system satisfy a second-order algebraic equation a 11 x 2 + a 22 y 2 + a 33 z 2 + 2a 12 xy + 2a 13 xz + 2a 23 yz + 2a 14 x + 2a 24 y + 2a 34 z + a 44 = 0,(4.7.2.1) or (a 11 x + a 12 y + a 13 z + a 14 )x +(a 21 x + a 22 y + a 23 z + a 24 )y +(a 31 x + a 32 y + a 33 z + a 34 )z + a 41 x + a 42 y + a 43 z + a 44 = 0, where a ij = a ji (i, j = 1, 2, 3, 4). If equation (4.7.2.1) does not define a real geometric object, then one says that this equation defines an imaginary quadric. Equation (4.7.2.1) in vector form reads (Ar) ⋅ r + 2a ⋅ r + a 44 = 0,(4.7.2.2) where A is the affinor with coordinates A i j = a ij and a is the vector with coordinates a i = a i4 . 4.7.2-2. Classification of quadrics. There exists a rectangular Cartesian coordinate system in which equation (4.7.2.1), depend- ing on the coefficients, has 1 of 17 canonical forms, each of which is associated with a certain class of quadrics (see Table 4.3). 4.7.2-3. Invariants of quadrics. The shape of a quadric can be determined by using four invariants and two semi-invariants without reducing equation (4.7.2.1) to canonical form. The main invariants are the quantities S = a 11 + a 22 + a 33 ,(4.7.2.3) T =    a 11 a 12 a 21 a 22    +    a 11 a 13 a 31 a 33    +    a 22 a 23 a 23 a 33    ,(4.7.2.4) δ =      a 11 a 12 a 13 a 12 a 22 a 23 a 13 a 23 a 33      ,(4.7.2.5) Δ =        a 11 a 12 a 13 a 14 a 12 a 22 a 23 a 24 a 13 a 23 a 33 a 34 a 14 a 24 a 34 a 44        ,(4.7.2.6) whose values are preserved under parallel translations and rotations of the coordinate axes. 4.7. QUADRIC SURFACES (QUADRICS) 149 TABLE 4.3 Canonical equations and classes of quadrics No. Surface Canonical equation Type Class Irreducible surfaces 1 Ellipsoid x 2 a 2 + y 2 b 2 + z 2 c 2 = 1 Elliptic 2 Imaginary ellipsoid x 2 a 2 + y 2 b 2 + z 2 c 2 =–1 3 One-sheeted hyperboloid x 2 a 2 + y 2 b 2 – z 2 c 2 = 1 Hyperbolic Nondegenerate 4 Two-sheeted hyperboloid x 2 a 2 + y 2 b 2 – z 2 c 2 =–1 5 Elliptic paraboloid x 2 p + y 2 q = 2z Parabolic (p > 0, q > 0) 6 Hyperbolic paraboloid x 2 p – y 2 q = 2z 7 Elliptic cylinder x 2 a 2 + y 2 b 2 = 1 8 Imaginary elliptic cylinder x 2 a 2 + y 2 b 2 =–1 Cylindrical 9 Hyperbolic cylinder x 2 a 2 – y 2 b 2 = 1 Degenerate 10 Parabolic cylinder y 2 = 2px 11 Real cone x 2 a 2 + y 2 b 2 – z 2 c 2 = 0 Conic 12 Imaginary cone with real vertex x 2 a 2 + y 2 b 2 + z 2 c 2 = 0 Reducible surfaces 13 Pair of real intersecting planes x 2 a 2 – y 2 b 2 = 0 14 Pair of imaginary planes intersecting in a real straight line x 2 a 2 + y 2 b 2 = 0 Pairs of planes Degenerate 15 Pair of real parallel planes x 2 = a 2 16 Pair of imaginary parallel planes x 2 =–a 2 17 Pair of real coinciding planes x 2 = 0 The semi-invariants are the quantities σ = Δ 11 + Δ 22 + Δ 33 ,(4.7.2.7) Σ =    a 11 a 14 a 41 a 44    +    a 22 a 24 a 42 a 44    +    a 33 a 34 a 44 a 44    ,(4.7.2.8) whose values are preserved only under rotations of the coordinate axes. Here Δ ij is the cofactor of the entry a ij in Δ. The classification of quadrics based on the invariants S, T , δ,andΔ and the semi- invariants σ and Σ is given in Tables 4.4 and 4.5. . + y b  . (4.7.1.7) One of these families is shown in Fig. 4.56a. Y Y ZZ ()a ()b X X O O Figure 4.56. Families of rulings for one-sheeted hyperboloid (a) and for hyperbolic paraboloid (b). Properties of rulings of. under rotations of the coordinate axes. Here Δ ij is the cofactor of the entry a ij in Δ. The classification of quadrics based on the invariants S, T , δ ,and and the semi- invariants σ and Σ is given. the coefficients, has 1 of 17 canonical forms, each of which is associated with a certain class of quadrics (see Table 4.3). 4.7.2-3. Invariants of quadrics. The shape of a quadric can be determined