Derivation of equations for diametral planes and centers

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DERIVATION OF EQUATIONS FOR DIAMETRAL PLANES AND CENTERS

Intersection of a line and a quadric surface. Given: A line L passing through point P₀(x₀, y₀, z₀) in direction (λ, μ, ν) expressed in parametric form as

x = x₀ + λt

1) y = y₀ + μt

z = z₀ + νt

and the quadric surface Q given by

2) Q(x, y, z) = ax² + by² + cz² + 2fyz + 2gxz + 2hxy + 2px + 2qy + 2rz + d = 0

(where λ, μ, ν are direction cosines). The points, real or imaginary, at which line L intersects quadric surface Q correspond to the roots of the following equation in t:

3) Q(x₀ + λt, y₀ + μt, z₀ + νt ) = 0 .

Upon expansion this equation becomes

4) Q(x₀ + λt, y₀ + μt, z₀ + νt ) = e(λ, μ, ν)t² + 2Φ(x₀, y₀, z₀, λ, μ, ν)t + Q(x₀, y₀, z₀) = 0

where

5) e(λ, μ, ν) = aλ² + bμ² + cν² + 2fμν + 2gλν + 2hλμ

= aλλ + hλμ + gλν

+ hμλ + bμμ + fμν

+ gνλ + fνμ + cνν

= (aλ + hμ + gν)λ + (hλ + bμ + fν)μ + (gλ + fμ + cυ)ν

and

6) Φ(x₀, y₀, z₀, λ, μ, ν) = (ax₀ + hy₀ + gz₀ + p)λ + (hx₀ + by₀ + fz₀ + q)μ + (gx₀ + fy₀ + cz₀ + r)ν

= (aλ + hμ + gν)x₀ + (hλ + bμ + fν)y₀ + (gλ + fμ + cν)z₀ + (pλ + qμ + rν)

There are four cases to consider:

(i) e(λ, μ, ν) ≠ 0 . The equation 4) is a quadratic equation with real coefficients, whose roots are either real or conjugate imaginary numbers. The line L meets the surface Q in two points, real or imaginary, which correspond to the two roots of 4). The line is said to determine a chord of the quartic surface, even if the two points of intersection are coincident. The midpoint of this chord corresponds to the arithmetic mean of the two roots of 4), and is always real.

(ii) e(λ, μ, ν) = 0, Φ(x₀, y₀, z₀, λ, μ, ν) ≠ 0 . The equation 4) is linear, and line L meets surface Q in one real point.

(iii) e(λ, μ, ν) = 0, Φ(x₀, y₀, z₀, λ, μ, ν) = 0, Q(x₀, y₀, z₀) ≠ 0 . The line L and the quadric surface have no points in common, real or imaginary.

(iv) e(λ, μ, ν) = 0, Φ(x₀, y₀, z₀, λ, μ, ν) = 0, Q(x₀, y₀, z₀) = 0 . The line L lies entirely in the quadric surface, and is called a ruling of the surface.

Diametral planes. A diametral plane corresponds to the locus of the midpoints of the parallel chords created by a system of parallel lines cutting through a quadric surface in some specified direction. The equation of the diametral plane conjugate to a system of parallel chords cutting through a quadric surface

f(x, y, z) = ax² + by² + cz² + 2fyz + 2gxz + 2hxy + 2px + 2qy + 2rz + d = 0

in the direction (l, m, n) is:

(al + hm + gn)x + (hl + bm + fn)y + (gl + fm + cn) z + (pl + qm + rn) = 0

Derivation. Let line L be of a direction (λ, μ, ν) for which e(λ, μ, ν) ≠ 0 i.e. of a direction such that it intersects quadric surface Q in two points, real or imaginary, giving two roots, real or imaginary, for equation 4). The midpoint of these two points of intersection will be real. Let this midpoint be the fixed point P₀(x₀, y₀, z₀) referred to in equations 1) above. The sum of the two roots is zero. Why? Because these roots (at least if they are real) are equal in magnitude and opposite in sign. Conversely, if the sum of these roots is zero, P₀ is the midpoint of the chord. Now we observe that equation 4) is a quadratic equation of type ax² + bx + c = 0 with solution given by the quadratic formula

and a necessary and sufficient condition for the sum of the roots of such an equation to vanish is that b = 0. Therefore a necessary and sufficient condition for P₀ to be the midpoint of the chord under consideration is that Φ(x₀, y₀, z₀, λ, μ, ν) = 0. Thus the locus of midpoints of the chords of the quadric surface Q(x, y, z) = 0 with direction cosines λ, μ, ν is the plane

(aλ + hμ + gν)x + (hλ + bμ + fν)y + (gλ + fμ + cν)z + (pλ + qμ + rν) = 0 .

In this equation the direction cosines λ, μ, ν can be replaced by any set of direction numbers l, m, n of the family of chords giving

(al + hm + gn)x + (hl + bm + fn)y + (gl + fm + cn) z + (pl + qm + rn) = 0

Center. A center of a quadric surface is defined as follows: A center of a quadric surface is a point P_c with the property that any line through P_c

(i) determines a chord of the surface whose midpoint is P_c

(ii) has no point in common with the surface,

(iii) lies entirely in the surface.

Equations giving the center. The centers of the quadric surface Q(x, y, z) = 0 are the points whose coordinates are solutions of the system

ax + hy + gz + p = 0

hx + by + fz + q = 0

gx + fy + cz + r = 0 .

Derivation. The point P₀(x₀, y₀, z₀) is a center of the quadric surface Q if and only if Φ(x₀, y₀, z₀, λ, μ, ν) = 0 for every set of direction cosines λ, μ, ν; that is if and only if in

Φ(x₀, y₀, z₀, λ, μ, ν) = (ax₀ + hy₀ + gz₀ + p)λ + (hx₀ + by₀ + fz₀ + q)μ + (gx₀ + fy₀ + cz₀ + r)ν

the coefficients of λ, μ, and ν vanish. Thus we have the theorem:

The centers of the quadric surface Q(x, y, z) = 0 are the points whose coordinates are solutions of the system

ax + hy + gz + p = 0

hx + by + fz + q = 0

gx + fy + cz + r = 0 .