Chapter VII. Classification of Quadric Surfaces 65. Intersection of a quadric and a line. General form and its matrix representation.

–Intersection of a given line –With the quadric form, substitute the value from (2), we get the following form: –Let

–Then the related quantities are in matrix form

–The roots in r of equation (3) are the distances from point on line (2) to the points in which this line intersects the quadric. (parametric form is useful) –If, Eq. (3) is a quadratic in r. –If Q=0, but R and S are not both zero, (3) was a quadratic, with one or more infinite roots. –If Q=R=S=0, (2) is satisfied for all values of r.

–Theorem I. Every line which does not lie on a given quadric surface has two (distinct or coincident) points in common with the surface. –Theorem II. If a given line has more than two points in common with a given quadric, it lies entirely on the quadric.

66. Diametral planes, center. Let, be the intersection points of line (2) with the quadric. The segment is called a chord of the quadric. –Theorem I. The locus of the middle point of a system of parallel chords of a quadric is a plane. (* if it’s real, more accurately, on a plane, or the intersection of a plane with the quadric)

–Proof. Let be root of (3), then the condition that is the middle point of the chord is that

–Equation (5) is linear in, and in fact a plane equation. It is called the diametral plane. –Theorem II. All the diametral planes of a quadric have at least one (finite or infinite) point in common. –Proof. For any unit vector V0, the plane (5) passes through the intersection of the planes

–Let G be the coefficient matrix, and D=|G|, the determinant. –If, the plane (6) intersect in a single finite point. If this point does not lie on the surface, it is called the center of the quadric. Otherwise, it is called a vertex of the quadric. –If D=0, but L,M,N are not all be zero, the plane (6) intersect in a single infinitely distant point.

–Exercises. P.77, No.1, 2,6,8. Equation of a quadric referred to its center. –If a quadric has a center, its equation, referred to its center as origin, may be obtained in the following way.

–Since is the center, it satisfies equation (6), and it means that the linear part in quadric disappears. We thus have: –By eliminating, we have

Therefore, DS=|A| (1) Or, if, S=det A/D, when det A=0, from (9), the quadric is a cone. (2) If D=0 and, then we have D=0. Since the center was assumed to be a finite point, so that it follows that L=M=N=0, and the surface has a line or plane of centers.

– is called the discriminant of the given quadric. If, the quadric is said to be singular, otherwise, non-singular. 68. Principal Planes. –A diametral plane which is perpendicular to the chords it bisects –Theorem. If the coefficient of a quadric are real, and if the quadric does not have the plane at infinity as a component, the quadric has at least one real, finite, principal plane.

–The condition that the diametral plane (5) –Is perpendicular to the chords it bisects is that the normal of plane (5) coincides with the direction cosine of the chord.

–The condition that these linear equation has none zero solution about direction cosine is its determinant of coefficient matrix is zero. –Thus k is the characteristic eigenvalue of matrix D

69. Reality of the roots of the discriminant. –Theorem I. The roots of the discriminanting cubic are real. –Notice: the coefficient matrix is real and symmetric, from linear algebra, it has “three real roots, countering on the multiple roots.”

–Theorem II. Not all the roots of the discriminating cubic are equal to zero. –* if they are all zero, D is then equivalent to a zero matrix, implies that a=b=c=f=g=h= Simplification of the equation of a quadric. –Let the axes be transformed in such a way that a real, finite principal plane of quadric F(x,y,z)=0 is taken as x=0.

–Then the surface is symmetric with respect to x=0, the coefficients of terms of first degree in x must all be zero. Hence the equation has the form:

–Moreover,, since otherwise x=0 would not be a principal plane. –Now let the planes y=0,z=0 be rotated about X- axis through the angle defined by –This rotation reduces the coefficient of yz to zero., now the equation has the form:

71. Classification of quadric surfaces. –Since the equation of a quadric can be always be reduced to the form of (16), a completed classification can be made –By considering the possible values of the coefficients. Here

–I. Let both b’ and c’ be different from zero. By a translation of the axes in such a way that –Is the new origin, the equation reduces to:

–II. Let –(1) if, by a translation of axes, the equation may be reduced to –This equation takes the form

–(2) if n’=0, this may be reduced to

–III. Let b’=c’=0. The equation (16) is in this case –(1) if m’ and n’ are not both zero, since the plane 2m’y+2n’z+d’=0 is perpendicular to x=0, we may rotate and translate the axes so that the plane is now y=0. The equation becomes

–(2) if m’ and n’ are both zero, we have

72. Invariants under motion. –Invariant under motion A function of the coefficients of the equation of a surface, the value of which is unchanged when the axes are rotated and translated. –E.g. I=a+b+c,

73. Proof that I,J, and D are invariant. –*hint. Using matrix representation, the result can be obtained much easily. –I, J, D is exactly the coefficients of the equation of eigenvalues. It is unchanged under motion.

74. Proof that is invariant –Using matrix representation. –Let U be a transformation, X=UX’, then –the coefficient matrix becomes

Exercises: P.89, No. 2,4,6,8,18.