Chapter 3. Quadric hypersurfaces. 3.1 Quadric hypersurfaces Denition.

Transcription

1 Chapter 3 Quadric hypersurfaces 3.1 Quadric hypersurfaces Denition. Denition 1. In an n-dimensional ane space A; given an ane frame fo;! e i g: A quadric hypersurface in A is a set S consisting all points whose coordinates in the given frame are zeros of a quadric polynomial, i.e. they satisfy the following quadric equation: a ij x i x j + 2 a i x i + a 0 = 0; (3.1) where a ij 2 K are not all zeros and a ij = a ji ; 8i; j = 1; 2; : : : ; n: This means that the matrix is symmetric and non-zero. A = 0 a 11 a 12 : : : a 1n a 21 a 22 : : : a 2n a n1 a n2 : : : a nn 1 C A The equation (3.1) is called the equation of the quadric hypersurface in the given frame and can be written in the matrix form [x] t A[x] + 2 [a] t [x] + a 0 = 0: (3.2) Usually, a quadric hypersurface in A 2 (K) is called a quadratic curve and a quadric hypersurface in A 3 (K) is called a quadric surface. 45

2 Let B = 0 Ane and Euclidean Geometry (version1) 1 a 0 a 1 a 2 : : : a n a 1 a 11 a 12 : : : a 1n a 2 a 21 a 22 : : : a 2n C A : a n a n1 a n2 : : : a nn A quadric hypersurface is said to be nondegenerate (or nonsingular) if B is nondegenerate, i.e. invertible (det B 6= 0) and is said to be degenerate (or singular) if B is degenerate, i.e. det B = 0. For a degenerate quadric hypersurface, if rank A = rank B; then it is called a quadric hypercone and if rank A 6= rank B; then it is called a quadric hypercylinder. The meaning of the denitions can be seen in Exercises 3.19 and The denition of a quadric hypersurface is well-dened, i.e. does not depend on the given frame. Indeed, suppose that we have a change of coordinates [x] = C[x 0 ] + [b]: Substitute the value of [x] into equation (3.2), we get [x] t A[x] + 2 [a] t [x] + a 0 = 0, (C[x 0 ] + [b]) t A(C[x 0 ] + [b]) + 2[a] t (C[x 0 ] + [b]) + a 0 = 0, ([x 0 ] t C t + [b] t )A(C[x 0 ] + [b]) + 2[a] t (C[x 0 ] + [b]) + a 0 = 0, [x 0 ] t C t AC[x 0 ] + ([x 0 ] t C t A[b] + [b] t AC[x 0 ] + 2[a] t C[x 0 ]) + [b] t A[b] + 2[a] t [b] + a 0 = 0: Since [x 0 ] t C t A[b] and [b] t AC[x 0 ] are squared matrix of order 1, Therefore, (3.3) becomes ([x 0 ] t C t A[b]) t = [b] t AC[x 0 ] = [x 0 ] t C t A[b]: (3.3) [x 0 ] t C t AC[x 0 ] + 2([b] t AC + 2[a] t C)[x 0 ] + [b] t A[b] + 2[a] t [b] + a 0 = 0: (3.4) 2. It is easy to prove that rank A and rank B do not depend on the given frame. Example 1. In A 2 ; the equation x 2 a 2 determines a quadratic curve called an ellipse. Example 2. In A 3 ; the equation x 2 a 2 + y2 b 2 determines a quadratic curve called an ellipsoid. + y2 b 2 = 1; a > 0; b > 0; + z2 c 2 = 1; a > 0; b > 0; c > 0; 46

3 Theorem If S is a quadric hypersurface in A n and f is an ane automorphism of A n ; then f(s) is a quadric hypersurface in A n : Proof. Suppose that (3.1) is the equation of S in a given ane frame fo;! e i g and f : A n! A n is an ane automorphism. Let fo 0 ;! w i g is the image of fo;! e i g under f; i.e. O 0 = f(o) and! wi =! f (! e i ); i = 1; 2; : : : ; n: Suppose that (x 1 ; x 2 ; : : : ; x n ) is the coordinates of a point M in the frame fo;! e i g: Since! O 0 f(m) =!! f ( OM) =! f ( x i! ei ) = x i! f (! ei ) = x i! wi ; (x 1 ; x 2 ; : : : ; x n ) is the coordinates of the point M 0 = f(m) in the frame fo 0 ;! w i g: This proves that, the coordinates (in the frame fo 0 ;! w i g) of a point in f(s) satises the equation (3.1). In other words, the equation of S in the frame fo;! e i g and the equation of f(s) in the frame fo 0 ;! w i g are the same. By the denition, f(s) is a quadric hypersurface. Of course, singulatity and nonsingularity of a quadric hypersurface are preserved Center of a quadric hypersurface. Denition 2. The center of a quadric hypersurface S is a point I such that if I is chosen as the origin, then the equation of S is of the simple form [x] t A[x] + a = 0; (3.5) If the center I of S belongs to S; then I is called a singular point of S: A quadric hypersurface S is symmetric about its center. This means that the symmetry about I; i.e. the dilation with center I; and ratio 1 of A n maps S into itself. Indeed, the coordinates (x 1 ; : : : ; x n ) of a point M satisfy (3.5) if and only if the coordinates ( x 1 ; : : : ; x n ) of M 0 ; the symmetric point of M about I; satisfy the equation (3.5). That means S is symmetric about I: The opposite is also true. If a quadric hypersurface S is symmetric about a point I; then I is a center of S (see Exercise 3.18). Theorem In an ane space A n ; let S be a quadric hypersurface and (3.2) be its equation in a given frame fo;! e i g: Then S has a unique center if and only if det A 6= 0: If det A = 0; then S has innitely centers or it has no center. Proof. Suppose that equation of a quadric hypersurface S is (3.2) and I is a point whose coordinates are (b 1 ; b 2 ; : : : ; b n ) in the frame fo;! e i g: The formula of the change of coordinates from fo;! e i g to fi;! e i g is [x] = [x 0 ] + [b]: 47

4 Substitute [x] into Equation (3.2) we obtain an equation of S in new frame fi;! e i g is [x 0 ] t A[x 0 ] + 2(A[b] + [a]) t [x 0 ] + [b] t A[b] + 2[a] t [b] + a 0 = 0: By the denition, I is a center of S if and only if A[b] + [a] = 0: In other words, I is a center of S if and only if its coordinates is a solution of the system of equations A[x] + [a] = 0: (3.6) Thus, S has only one center if and only if Equation (3.6) has only one solution, i.e. det A 6= 0: If det A = 0; then (3.6) has no solutions (S has no centers) or it has innitely many solutions(s has innitely many centers). Remark By the proof of Theorem 3.1.1, an ane automorphism f maps a quadric hypersurface S with a center I to a quadric hypersurface f(s) with a center f(i): 2. By the proof of Theorem 3.1.2, if rank A = r n and Equation (3.6) has at least one solution, then the set of all solutions of (3.6) determined an (n r)-plane in A n : 3. By the proof of Theorem 3.1.2, for nding centers of a quadric hypersurface, we have to solve the system (3.6). 4. For nding singular points we have to solve the system ( A[x] + [a] = 0 [x] t A[x] + 2[a] t [x] + a 0 = 0 or have to solve the equivalent system ( A[x] + [a] = 0 [a] t [x] + a 0 = 0 : Asymptotic directions and asymptotes Denition Let S be a quadric hypersurface whose equation in a given frame fo;! e 1 ;! e 2 ; : : : ;! e n g is (3.1). A vector! c 6=! 0 is called an asymptotic vector if its coordinates (c 1 ; c 2 ; : : : ; c n ) in the basis f! e 1 ;! e 2 ; : : : ;! e n g of! A n satisfy the following condition or in the matrix form a ij c i c j = 0; [c] t A[c] = 0: The direction of an asymptotic vector, i.e. the 1-dimensional vector subspace h! c i generated by! c 6=! 0 is call an asymptotic direction. 48

5 2. If S has only one center and has an asymptotic vector! c ; then the line passing through the center with the directional vector! c is called an asymptote of the quadric hypersurface S: Then the set of all asymptotes of S forms a hypercone called the asymptotic hypercone of S (see Exercise 3.19). 3. Given two non-zero vectors in V n ;! c (c 1 ; :::; c n ); and! d (d 1 ; :::; d n ): Then! c is said to be conjugate to! d ; or the direction h! c i is conjugate to the direction h! d i w.r.t. S if a ij c i d j = 0; or in the matrix form [c] t A[d] = 0: 4. A direction h! c i is called a special direction of S if! c is conjugate (w.r.t. S) to every non-zero vector. Remark By the denition and the proof of Theorem we can see that an ane automorphism f maps an asymptote to an asymptote and! f maps an asymptotic vector of S to a such one of f(s): Of course, two conjugate directions (of S) are mapped to two conjugate directions (of f(s)). 2. If! c is conjugate to! d ; then! d is conjugate to! c (w.r.t. S). 3. A vector! c 6=! 0 is an asymptotic vector of S if and only if! c is conjugate to itself (w.r.t. S). 4. Every special direction is an asymptotic one. Example 3. In A 2 - an ellipse x2 a + y2 2 b 2 = 1 has no asymptotic directions. - a hyperbola x2 y 2 = 1 has two asymptotic directions, that are! c a 2 b 2 1 (a; b) and! c 2 (a; b): The corresponding asymptotes are y = b a x and y = b a x: - a parabola y 2 = 2px has an asymptotic vector! c (0; 1); but has no asymptote because it has no centers Diametral hyperplanes. The intersection between a line and a quadric hypersurface. Let d be a line passing through a point B(b 1 ; b 2 ; : : : ; b n ) and having! c (c 1 ; c 2 ; : : : ; c n ) as a directional vector. Then a parametric equation of d is x i = c i t + b i i = 1; 2; : : : ; n; (3.7) 49

6 or in the matrix form [x] = [c]t + [b]: (3.8) The coordinates of the intersection points between d and a quadric hypersurface S that has Equation (3.2) is a solution of a system consisting (3.2) and (3.8). Substitute the value of [x] in (3.8) into (3.2) we obtain the equation Expand this equation, we get where We have the following cases: ([c] t t + [b] t )A([c]t + [b]) + 2[a] t ([c]t + [b]) + a 0 = 0: [c] t A[c]t 2 + 2P t + Q = 0; (3.9) P = [b] t A[c] + [a] t [c] = Q = a ij b i b j + 2 a ij b i c j + a i b i + a 0 : a i c i ; 1. Vector! c is not an asymptotic vector, i.e. [c] t A[c] 6= 0: Then because (3.9) is a quadric equation, it may have two distinct real roots; two complex roots which are complex conjugates of each other or a double root. Correspondingly, the line intersects the quadric hypersurface at two points, does not intersect the quadric hypersurface (in this case we can think that the line intersects the quadric hypersurface at two imaginary points) or intersect the quadric hypersurface at a double points M 0 (in this case we say that the line is tangent to the quadric hypersurface at M 0 ). 2. Vector! c is an asymptotic vector, i.e. [c] t A[c] = 0: Equation (3.9) becomes 2P t + Q = 0; (3.10) (a) If P 6= 0; then Equation (3.9) has only one root. The line intersect the quadric hypersurface at one point. (b) If P = 0 and Q 6= 0; then Equation (3.9) has no roots. The line does not intersect the quadric hypersurface. (c) If P = Q = 0; then every value of t is a root of (3.9). Therefore, the line is a subset of the quadric hypersurface. Diametral hyperplanes. In A n ; let S be a quadric hypersurface that has equation (3.2) in a given frame fo;! e i g: Consider the set of parallel lines that have! c (c 1 ; :::; c n ) as their directional vector. Suppose that! c is not an asymptotic vector of S: Let d be a line in this set. Suppose that d intersects S at two real points that are distinct or coincident M 1 and M 2. Let B(b 1 ; b 2 ; : : : ; b n ) be the midpoint of the segment M 1 M 2 : Then d has Equation (3.8), and M 1 ; M 2 corresponding to two values t 1 ; t 2 ; that are distinct or coincident. 50

7 Their coordinates are two real roots of the equation (3.9) and can be written in the matrix form [b] + [c]t 1 and [b] + [c]t 2 : The segment M 1 M 2 is called a chord of S: Since B is the midpoint of the chord M 1 M 2 (i.e. Therefore, t 1 = t 2 :! BM 1 = [c](t 1 + t 2 ) = 0:! BM 2 ) we get The condition for the sum of two roots of Equation (3.9) to be zero is P = [b] t A[c] + [a] t [c] = 0: Thus, if B is the midpoint of M 1 M 2 ; then it coordinates must satisfy the following equation [x] t A[c] + [a] t [c] = 0; or [c] t A[x] + [c] t [a] = 0: (3.11) Since! c is not an asymptotic vector, [c] t A 6= 0: Equation (3.11) determines a hyperplane. Any center of S (if exists) is on this hyperplane. The hyperplane dened by Equation (3.11) is called the diametral hyperplane of S conjugate to the direction h! c i of the chords. Thus, we have the following. Theorem The set of midpoints of parallel chords, whose directional direction is not asymptotic, of a quadric hypersurface S is a subset of a hyperplane. Example 4. In A 2 a diametral hyperplane is called a diametral line while in A 3 a diametral hyperplane is called a diametral plane. We have 1. The diametral line conjugate to the direction h! c (c 1 ; c 2 )i; of the ellipse x2 a + y2 2 b 2 line c 1 x a + c 2y = 0: 2 b 2 = 1 is the 2. The diametral line conjugate to the direction h! c (c 1 ; c 2 )i; of the hyperbola x2 y 2 a 2 b 2 the line c 1 x c 2 y = 0: a 2 b 2 = 1 is 3. The diametral line conjugate to the direction h! c (c 1 ; c 2 )i; of the parabola y 2 = 2px is the line c 2 y = pc 1 : 51

8 3.1.5 Tangent lines and tangent hyperplanes of a quadric hypersurface. Denition 4. In A n given a quadric hypersurface S: A line d is called a tangent line of S if: 1. either the direction of d is not an asymptotic one and d intersect S at a double point; in this case, d is said to be tangent to S at that point 2. or the direction of d is not an asymptotic one and the line d containing in S; in this case, d is said to be tangent to S at every point of d. Let S be a quadric hypersurface whose equation is (3.1), B 2 S is a point whose coodinates is (b 1 ; b 2 ; : : : ; b n );! c be a vector whose coordinates is (c 1 ; c 2 ; : : : ; c n ) and d be the line passing through B with directional vector! c : We will nd a necessary and sucient condition for d being a tangent line of S: Recall that, equation for determining intersection points between d and S is: where [c] t A[c]t 2 + 2P t + Q = 0; P = [b] t A[c] + [a] t [c] = Q = Since B 2 S; Q = 0: The equation becomes a ij b i b j + 2 a ij b i c j + [c] t A[c]t 2 + 2P t = 0; a i b i + a 0 : a i c i ; or ([c] t A[c]t + 2P )t = 0: (3.12) 1. If [c] t A[c] 6= 0; i.e! c is not an asymptotic vector, then d is a tangent line if and only if (3.12) has a double root, i.e. P = [b] t A[c] + [a] t [c] = 0: 2. If [c] t A[c] = 0; i.e.! c is an asymptotic vector, then d is a tangent line if and only if d S: This is equivalent to P = [b] t A[c] + [a] t [c] = 0: In other words, the line d is a tangent line if and only if [b] t A[c] + [a] t [c] = 0: (3.13) If B is a singular point of S; then every line passing through B is a tangent line of S at B: Indeed, if B is singular, then B is a center, therefore its coordinates satises the equation A[b] + [a] = 0; i.e. its coordinates satises (3.13), too. If B is not a singular point of S; then tangent lines of S at B form a hyperplane containing B: This hyperplane is called a tangent hyperplane of S at B: 52

9 ! Indeed, a point M(x 1 ; x 2 ; : : : ; x n ) lying on a tangent line passing through B if and only if BM =! c satises (3.13), i.e. [b] t A([x] [b]) + [a] t ([x] [b]) = 0: or ([b] t A + [a] t )([x] [b]) = 0: (3.14) Since B is not singular, [b] t A + [a] t = (A[b] + [a]) t 6= 0: Equation (3.14) determines a hyperplane, the tangent hyperplane of S at B: Since B 2 S; [b] t A[b] [a] t [b] = [a] t [b] + a 0 : Substitute this into (3.14), the equation of the tangent hyperplane of S at B becomes Example 5. In A 2 ; [b] t A[x] + [a] t ([x] + [b]) + a 0 = 0: (3.15) 1. an equation of the tangent line of the ellipse x2 a + y2 2 b 2 x 0 x a 2 + y 0y b 2 = 1; = 1 at M 0 (x 0 ; y 0 ) is 2. an equation of the tangent line of the hyperbola x2 y 2 a 2 b 2 x 0 x a 2 y 0 y b 2 = 1; = 1 at M 0 (x 0 ; y 0 ) is 3. an equation of the tangent line of the parabola y 2 = 2px at M 0 (x 0 ; y 0 ) is y 0 y = p(x + x 0 ): 3.2 The classication of quadric hypersurfaces Canonical equation of a quadric hypersurface. Theorem There exists a suitable ane frame such that a quadric hypersurface in A n has an equation of one of the following forms: 1. + : : : + x 2 k x2 k+1 : : : x2 r 1 = 0 (type I); 2. + : : : + x 2 k x2 k+1 : : : x2 r = 0 (type II); 3. + : : : + x 2 k x2 k+1 : : : x2 r 2x r+1 = 0 (typeiii). Equations of types I, II and III as above are called canonical equations of quadric hypersurfaces. 53

10 Proof. Suppose that an equation of S is (3.1). Consider the corresponding quadratic form H(! x ) = a ij x i x j : By results in Linear Algebra, we can nd a change of coordinates in! A n such that H has a canonical form. H(! x 0 ) = kx x 0 2 i Then the equation of S in the new frame is kx x 0 2 i rx i=k+1 rx i=k+1 x 0 i2 + 2 x 0 i2 ; 0 k n; 1 r n: a 0 ix 0 i + a 0 0 = 0: Under the change of coordinates 8 >< >: x 0 i = x i a 0 i x 0 i = x i + a 0 i x 0 i = x i i = 1; 2; : : : ; k i = k + 1; k + 2; : : : ; r i = r + 1; : : : ; n the above equation becomes We have the following cases: kx x 2 i rx x 2 i + 2 i=k+1 i=r+1 a 0 ix i + b = 0: (3.16) 1. The case of a 0 r+1 = : : : = a 0 n = 0 or r = n and b 6= 0: Under the change of coordinates 8 < : X i = X i = q Equation (3.16) is reduced the one of type I. 1 b i q b < 0 1 b i b > 0 2. The case of a 0 r+1 = : : : = a 0 n = 0 and b = 0: Equation (3.16) is of type II. 3. The case of there exists a 0 j 6= 0 (j > r): We can suppose that a 0 r+1 6= 0: Under the change of coordinates 8 < : X i = x i if i 6= r + 1 n X r+1 = a 0 b j x j j=r+1 2 Equation (3.16) is reduced to the one of type III. 54

11 Example 6. In A 3 ; let S be a quadric hypersurface whose equation in a given frame fo;! e 1 ;! e 2 ;! e 3 g is x 1 x 2 + 2x 1 x 3 + 2x 1 + 6x = 0: Find a canonical equation of S and corresponding frame. Solution. Consider the corresponding quadratic form in! A 3 We have Under the change of coordinates H(! x ) = x 1 x 2 + 2x 1 x 3 : H(! x ) = (x 1 + x 2 + x 3 ) 2 + (x 2 x 3 ) 2 : 8 >< >: X 1 = x 1 + x 2 + x 3 X 2 = x 2 x 3 ; (3.17) X 3 = x 3 or equivalently, 8 > < >: H can be written in the canonical form x 1 = X 1 X 2 2X 3 x 2 = X 2 + X 3 ; (3.18) x 3 = X 3 H(! X ) = X X 2 2 X 2 3: The corresponding basis is f! w 1 ;! w 2 ;! w 3 g; where! w 1 =! e 1 ;! w 2 =! e1 +! e 2 ;! w 3 = 2! e 1 +! e 2 +! e 3 : We can consider (3.18) as a change of coordinates in A n : Then equation of S in the new frame fo;! w 1 ;! w 2 ;! w 3 g is X X 2 2 X X 1 + 4X 2 + 2X = 0: Under the change of coordinates 8 > < >: we obtain the canonical equation X 1 = p 2y 1 1 X 2 = p 2y 2 2 X 3 = p ; (3.19) 2y z z 2 2 z = 0; (3.20) and the corresponding frame fi;! u 1 ;! u 2 ;! u 3 g; where the coordinates of the point I is ( 1; 2; 1) in the given frame fo;! e 1 ;! e 2 ;! e 3 g and! u1 = p 2! e 1 ;! u2 = p 2(! e1 +! e 2 );! u3 = p 2( 2! e 1 +! e 2 +! e 3 ): Equation (3.20) is a canonical equation of S: 55

12 Remark If a canonical equation of S is of type I, then S has centers and the centers are not belonging to S: If a canonical equation of S is of type II, th S then S has centers and the centers are belonging to S: If a canonical equation of S is of type III, then S has no centers. 2. Since r is the rank, while k and (r k) are positive and negative indexies of the quadratic form, they are invariant under a change of coordinates. This means that, two equations that are of the same type but are dierent in r or k or r k can not be a canonical equation of a quadric hypersurface The classication of quadric hypersurfaces. Denition 5. Two quadric hypersurfaces in A n are called of the same kind if their canonical equations are of the same type and are of the same values r; k: In other words, two quadric hypersurfaces are of the sam kind if their canonical equations (in suitable frames) are the same. Theorem Two quadric hypersurfaces are anely equivalent, i.e. there is an ane automorphism that maps a quadric hypersurface to the other, if and only if they are of the same kind. In other words, the classication in Denition 5 is the ane classication. Proof. Suppose that the canonical equation of S in the frame fo;! e i g and the canonical equation of S 0 in the frame fo 0 ;! e i 0 g are the same. Consider the ane automorphism f that maps the frame fo;! e i g to the frame fo 0 ;! e i 0 g: We have f(s) = S 0 : Conversely, suppose that f is an ane automorphism and f(s) = S 0 : Let fo;! e i g be an ane frame such that the equation of S in this frame is canonical and fo 0 ;! e i 0 g is the image of fo;! e i g under f. Then the canonical equation of S 0 in the frame fo 0 ;! e i 0 g is the same as that of S in the frame fo;! e i g: Therefore, S and S 0 are of the same type. Below are tables of the classications of quadric hypersurfaces in A 2 and A 3 : The classication of quadratic curves in A 2 : In A 2 ; equation of a quadratic curve is of general form as follows a 11 + a a 12 x 1 x 2 + 2a 1 x 1 + 2a 2 x 2 + a = 0: By Theorems and 3.2.2, based on the types of their canonical equations, we can arrange quadratic curves together their names as follows 56

13 1. 1 = 0 ellipse; 2. 1 = 0 hyperbola; 3. 1 = 0 imaginary ellipse; 4. = 0 a pair of intersecting imaginary lines (at a real point); 5. = 0 a pair of intersecting lines; 6. 2x 2 = 0 parabola; 7. 1 = 0 a pair of parallel lines; 8. 1 = 0 a pair of parallel imaginary lines; 9. = 0 a pair of coincident lines. The classication of quadric surfaces in A 3 : In A 3 ; equation of a quadric surface is of general form as follows a 11 + a 22 + a 33 x a 12 x 1 x 2 + 2a 13 x 1 x 3 + 2a 23 x 2 x a 1 x 1 + 2a 2 x 2 + 2a 3 x 3 + a = 0: By Theorems and 3.2.2, based on the types of their canonical equations, we can arrange quadric surfaces together their names as follows 1. 1 = elipsoid; 2. x = 0 hyperboloid of 1-sheet ; 3. x = 0 hyperoloid of 2-sheets; 4. x = 0 imaginary ellipsoid; 5. = 0 imaginary cone; 6. x 2 3 = 0 cone; 7. 2x 3 = 0 paraboloid elliptic; 8. 2x 3 = 0 paraboloid hyperbolic; 9. 1 = 0 elliptical cylinder; = 0 hyperbolic cylinder; = 0 imaginary elliptical cylinder; 12. = 0 a pair of intersecting imaginary planes; 13. = 0 a pair of intersecting planes; = 0 a pair of parallel planes; = 0 a pair of parallel imaginary planes; 16. = 0 a pair of coincident planes; 17. 2x 2 = 0 parabolic cylinder. EXERCISES Exercise 3.1. In the table of the classication of quadratic curves in A 2, study the curves one by one: 1. Is the curve singular or not? What is the rank of matrix A and B? Does it have centers? 2. Find asymptotic directions and asymptotes (if they exist). Exercise 3.2. The same questions as in Exercise 3.1 but for the table of the classication of quadric surfaces in A 3 : 57

14 Exercise 3.3. Prove that in A n ; a nonsingular quadric hypersurface has no centers or has only one center. Exercise 3.4. Consider the relative position between a quadric hypersurface and an m-plane in A n : (Hint: Choose a suitable frame). Exercise 3.5. Prove that, an asymptote (if there is) of a nonsingular quadric hypersurface never intersect the quadric hypersurface. Exercise 3.6. Prove that, if a quadric hypersurface S has a singular point, then S is singular. Exercise 3.7. In A 2 with a given frame, consider the following quadratic curves 1. S 1 : 4x 12 + x x 1 x 2 + 2x 2 = 0: 2. S 2 : 2 3x 12 x 1 x 1 + 2x = 0: 3. S 3 : 2 x 1 4x x 1 x 2 + 2x 2 = 0: 4. S 4 : x 1 x 2 + 4x 1 6x 2 = 0: 5. S 5 : 2x 1 x 1 + 2x = 0: 6. S 6 : 4 4x 1 x 2 8x 1 + 4x = 0: 7. S 7 : 3 6x 1 x 2 + 2x 1 + 2x 2 = 0: Find their centers, singular points, asymptotic directions and asymptotes. Exercise 3.8. For quadric curves in Exercise 3.7, nd their canonical equations and corresponding ane frames. Exercise Let! = (1; 2): Find the diametral line conjugate to the direction h! i of the curves given in Exercises Let A(0; 0) 2 S 1 : Write an equation of the tangent line of S 1 at A: Let B(0; 1) =2 S 2 : Write an equation of the tangent line of S 2 passing through B: Exercise In A 3 given quadric surfaces whose equations in a given frame are: 1. S 1 : x x 22 + x x 1 x 2 + 6x 2 x 3 + 2x 1 x 3 2x 1 + 6x 2 + 2x 3 = 0: 2. S 2 : 2 x 1 2x 22 + x x 1 x 2 8x 1 x 3 14x 1 14x x = 0: 3. S 3 : x 12 + x 22 + x x 1 x 1 2x = 0: 4. S 4 : x x 1 x 2 + 2x 1 x 3 + 2x 2 x 3 + 2x 2 + 2x 3 = 0: 5. S 5 : x 1 x 2 + 4x 1 x 3 2x 2 x 3 4x 1 4 = 0: 6. S 6 : x 1 x 2 + 4x 1 x 2 + 6x 2 x 3 2x 2 + 2x 3 2 = 0: 7. S 7 : 2 2 x 1 x x 3 6x 1 x 3 4x 2 x 3 + x 1 + x 2 x 3 = 0: 58

15 Find their centers, singular points, asymptotic directions and asymptotic cones (see Exercise 3.21). Exercise For quadric surfaces in Exercise 3.10, nd their canonical equations and corresponding ane frames. Which one is singular? Exercise Let A( 4 3 ; 2 3 ; 1 3 ) 2 S 1: Write an equation of the tangent hyperplane of S 1 at A: 2. Choose a point B on each of the quadric surfaces in Exercises 3.10 and write an equation of the tangent hyperplane of the surface at B: Exercise Let! d = (1; 2; 1): Find the diametral hyperplane conjugate to the direction h! d i of the quadric surfaces given in Exercises Exercise In A 2 let d be a line whose equation is 2x 1 +3x 2 3 = 0: Consider the intersection between d and quadratic curves in Exercises 3.7. Exercise In A 3 with a given ane frame ffo;! e 1 ;! e 2 ;! e 3 g; let S be a quadric surface whose equation is 2 + 4x 1 x 2 8x 1 x 3 14(x 1 x 2 + x 3 ) + 17 = 0: 1. Find centers of S: 2. Prove that the vector! c (1; 2; 3) is not an asymptotic vector of S: Write an equation of the diametral hyperplane of S conjugate to h! c i: 3. Prove that M 0 (1; 1; 2) 2 S is not a singular point of S: Write an equation of the tangent hyperplane of S at M 0 : Exercise In A 3 with a given frame fo;! e 1 ;! e 2 ;! e 3 g; let S 1 and S 2 be quadric surfaces whose equations, respectively, are: and Are S 1 and S 2 anely equivalent? + 2x 1 (x 1 + x 2 + x 3 ) + 1 = x x 1 x 2 + 2x 1 x 3 + 2(3x 1 + 5x 2 + x 3 ) = 0: Exercise In A 3 with a given frame fo;! e 1 ;! e 2 ;! e 3 g; let S be a quadric surface whose equation is x 1 x 2 + 2x 1 x 3 + 2x 1 + 2x 2 = 0: 1. Find a canonical equation of S and the correspondent frame. 2. Prove that the set of all tangent lines passing through I(1; 0; 1) is a quadric surface and write an equation of that quadric surface. Exercise Prove that if I is a point of symmetry of a quadric hypersurface S; the it is a center of S: 59

16 Exercise Recall that, a singular quadric hypersurface in A n with rank A = rank B is called a hypercone. Then rank A(= rank B) is called the rank of hypercone. 1. Prove that an ane automorphism maps a hypercone of rank r to a such one. 2. Suppose that S is a hypercone of rank r: Prove that there exists an ane frame fo;! e i g in which the equation of S is where rank(a ij ) = r: rx a ij x i x j = 0; a ij = a ji ; 3. Let O be the origin of the frame in item 2, prove that if S is a hypercone and M 2 S; then the line OM is a subset of S: The line OM is called rullings of S: 4. Prove that, for a hypercone of rank r, the set of all singular points is an (n r)-plane : Prove that for every M 2 S n ; M + S: 5. Classify hypercones in A 2 ; A 3 : Exercise Recall that, a singular quadric hypersurface in A n with rank A 6= rank B is called a hypercylinder. 1. Prove that an ane automorphism maps a hypercylinder to a such one. 2. Suppose that S is a hypercylinder. Prove that there exists an ane frame fo;! e i g in which the equation of S is one of the followings: or rx rx a ij x i x j + a = 0; a 6= 0; a ij = a ji ; i; j = 1; : : : ; r; (3.21) a ij x i x j + 2a r+1 x r+1 = 0; a r+1 6= 0; a ij = a ji ; i; j = 1; : : : ; r: (3.22) 3. Prove that a hypercylinder has no singular points. 4. Let! be a vector subspace generated by f! e r+1 ; : : : ;! e n g: Prove that, if M 2 S then the ane set passing through M and directional space! is a subset of S: 5. Let! be a vector subspace generated by f! e 1 ; : : : ;! e r g: Prove that the intersection between S and r-plane passing through O (where O is the origin of the frame in item 2) and directional space! is a quadric hypersurface in whose equation in the frame fo;! e i g is just (3.21) or (3.22); respectively. This quadric hypersurface (in ) is called the base of hypercylinder, denoted by S 0 : 6. Prove that if : A! is the parallel projection onto with direction! ; then (S) = S 0 : 7. Classify hypercylinders in A 2 ; A 3 : 60

17 Exercise In A n, let S be a nonsingular quadric hypersurface that has a center and an asymptote. Prove that the set of all asymptotes of S passing through the center is a hypercone, called the asymptotic hypercone. Find the rank and an equation of that hypercone. Exercise Let S be a quadric hypersurface that does not contain any line and d be the tangent line of S at M 2 S: Prove that 1.! d is not an asymptotic vector, 2. M belongs to the diametral hyperplane conjugate to! d : Exercise In A 2 ; given quadratic curves whose equations in a given frame are: 1. S 1 : 4 + 4x 1 x x = 0: (Ellipse). 2. S 2 : 4 + 4x 1 x 2 2x = 0 (Parabola). 3. S 3 : + 4x 1 x = 0 (Hyperbola). 4. S 4 : 6x 1 x x 2 8 = 0 (a pair of intersecting lines). 5. S 5 : 2x 1 x 1 + 2x 2 3 = 0 (a pair of parallel lines). Find their centers, singular points, asymptotic directions and asymptotes. Exercise For quadric curves given in Exercise 3.23, write canonical equations and corresponding frame. Exercise In A 3 given quadric surfaces whose equations in a given frame are: 1. S 1 : 2x 1 x 2 + 2x 1 x x 2 x 3 + 6x = 0: 2. S 2 : 4 4x 1 x 2 8x 1 x x 2 x 3 4x = 0: 3. S 3 : 9 12x 1 x 2 + 6x 1 x x 2 x 3 x 2 3 4x 2 + 4x 3 5 = 0: 4. S 4 : + 6x 1 x 1 x x x 3 16 = 0: 5. S 5 : 8 20x 1 x 2 + 4x 1 x x 2 x 3 2 = 0: 6. S 6 : 6x 1 x 2 4x 1 x x 2 x 3 + 3x = 0: 7. S 7 : 6x 1 x x 2 x 3 2x 2 4x x 3 1 = 0: 8. S 8 : 2x 1 x 2 + 4x 1 x 3 + 2x 1 4x 2 x 3 2x 2 + 4x x 3 = 0: 9. S 9 : 4x 1 x 2 + 2x x = 0: 10. S 10 : 4 + 4x 1 x 2 + 4x 1 x 3 + 2x 2 x 3 2 = 0: 11. S 11 : 13 12x 1 x 2 6x 1 x x 2 x x = 0: 61

18 12. S 12 : 4 4x 1 x 2 4x 1 x x 2 x 3 3x x 2 8x 3 5 = 0: Find their centers, singular points, asymptotic directions and asymptotic cones (if they exist). Exercise For quadric surfaces given in Exercises 3.25, write canonical equations and corresponding frame. Which one is a cylinder? Which one is a cone? Exercise In A 3 let S be a quadric surface whose equation is + 6x 1 x 1 x 3 2x x 2 x 3 4x 2 + 5x x = 0 and d 1 ; d 2 ; d 3 be three lines whose equations, respectively, are ( x2 2x 3 1 = 0 d 1 : x = 0 ( x1 3x 2 + 2x = 0 d 2 : x 3 2 = 0 ( x1 3x 2 + x 3 = 0 d 3 : x 3 1 = 0 Consider the intersections between S and the lines d i ; i = 1; 2; 3: Exercise In A 3 let S be a quadric surface whose equation is x 2 3 4x 1 x 2 6x 1 x x 2 x 3 2x 1 8x 3 5 = 0 and d 1 ; d 2 ; d 3 be three lines whose equations are ( x1 + 2x 2 + 3x 3 = 0 d 1 : x = 0 ( x1 + 2x 2 + 3x 3 = 0 d 2 : x = 0 ( x1 + 2x 2 + 3x 3 = 0 d 3 : x 2 x = 0 Consider the intersections between S and three lines d i ; i = 1; 2; 3: Exercise In A 3 let S be a quadric surface whose equation is x 1 x 2 + 2x 1 x 3 + 6x 2 x 3 2x 1 + 6x 2 + 2x 3 = 0 and be a plane whose equation is 2x 1 x 2 + x 3 4 = 0: Consider the intersection between and S: Exercise In A 3 let S be a quadric surface whose equation is + 4x x 1 x 2 + 6x 1 x 3 + 2x 2 x 3 + 4x 1 + 2x x 3 2 = 0 62

19 and 1 ; 2 ; 2 ; be three planes whose equations are 1 :x 1 + x 2 + 3x 3 = 0; 2 :x 2 + 2x 3 1 = 0; 3 :x 1 + x 2 + 4x 3 1 = 0: Prove that 1 \ S is an ellipse (in 1 ); 2 \ S is a hyperbola (in 2 ); 3 \ S is a parabola (in 3 ). Exercise In A n given a quadric surface S determened by the equation + + x 2 k x 2 k+1 x 2 n 1 = 0 (0 k < n): Prove that: 1. If n < 2k; then S contains some m-plane, where m n k; 2. If n = 2k; then S contains some m-plane, where m n k 1; 3. if n > 2k; then S contains some m-plane, where m k 1: Exercise In A 3 with a given frame fo;! e 1 ;! e 2 ;! e 3 g let S be a quadric surface and be a plane whose equations, respectively, are and 2x 1 x = 0 x 1 + x 2 + x 3 3 = 0: Let S 1 = S \ and l M be a line passing through M 2 S 1 with direction h! e 3 i: Let C be the union of all l M ; M 2 S: 1. Prove that C is a cylinder. 2. Find the image of S 1 under the parallel projection with direction h! e 3 i onto the rst coordinate plane. 63