SolitaryRoad.com

Website owner: James Miller

[ Home ] [ Up ] [ Info ] [ Mail ]

DERIVATION OF EQUATIONS FOR DIAMETRAL PLANES AND CENTERS

Intersection of a line and a quadric surface. Given: A line L passing through
point P_{0}(x_{0}, y_{0}, z_{0}) in direction (λ, μ, ν) expressed in parametric form as

x = x_{0} + λt

1) y = y_{0} + μt

z = z_{0} + νt

and the quadric surface Q given by

2) Q(x, y, z) = ax^{2} + by^{2} + cz^{2} + 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(x_{0} + λt, y_{0} + μt, z_{0} + νt ) = 0 .

Upon expansion this equation becomes

4) Q(x_{0} + λt, y_{0} + μt, z_{0} + νt ) = e(λ, μ, ν)t^{2} + 2Φ(x_{0}, y_{0}, z_{0}, λ, μ, ν)t + Q(x_{0}, y_{0}, z_{0}) = 0

where

5) e(λ, μ, ν) = aλ^{2} + bμ^{2} + cν^{2} + 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(x_{0}, y_{0}, z_{0}, λ, μ, ν) = (ax_{0} + hy_{0} + gz_{0} + p)λ + (hx_{0} + by_{0} + fz_{0} + q)μ + (gx_{0} + fy_{0} + cz_{0} +
r)ν

= (aλ + hμ + gν)x_{0} + (hλ + bμ + fν)y_{0} + (gλ + fμ + cν)z_{0} + (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(x_{0}, y_{0}, z_{0}, 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(x_{0}, y_{0}, z_{0}, λ, μ, ν) = 0, Q(x_{0}, y_{0}, z_{0}) ≠ 0 . The line L and the quadric surface
have no points in common, real or imaginary.

(iv) e(λ, μ, ν) = 0, Φ(x_{0}, y_{0}, z_{0}, λ, μ, ν) = 0, Q(x_{0}, y_{0}, z_{0}) = 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) = ax^{2} + by^{2} + cz^{2} + 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 P_{0}(x_{0}, y_{0}, z_{0}) 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, P_{0} is the midpoint of the chord.
Now we observe that equation 4) is a quadratic equation of type ax^{2} + 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 P_{0} to be the midpoint of the chord
under consideration is that Φ(x_{0}, y_{0}, z_{0}, 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}

or

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

or

(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 P_{0}(x_{0}, y_{0}, z_{0}) is a center of the quadric surface Q if and only if Φ(x_{0}, y_{0}, z_{0},
λ, μ, ν) = 0 for every set of direction cosines λ, μ, ν; that is if and only if in

Φ(x_{0}, y_{0}, z_{0}, λ, μ, ν) = (ax_{0} + hy_{0} + gz_{0} + p)λ + (hx_{0} + by_{0} + fz_{0} + q)μ + (gx_{0} + fy_{0} + cz_{0} + 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 .

More from SolitaryRoad.com:

Jesus Christ and His Teachings

Way of enlightenment, wisdom, and understanding

America, a corrupt, depraved, shameless country

On integrity and the lack of it

The test of a person's Christianity is what he is

Ninety five percent of the problems that most people have come from personal foolishness

Liberalism, socialism and the modern welfare state

The desire to harm, a motivation for conduct

On Self-sufficient Country Living, Homesteading

Topically Arranged Proverbs, Precepts, Quotations. Common Sayings. Poor Richard's Almanac.

Theory on the Formation of Character

People are like radio tuners --- they pick out and listen to one wavelength and ignore the rest

Cause of Character Traits --- According to Aristotle

We are what we eat --- living under the discipline of a diet

Avoiding problems and trouble in life

Role of habit in formation of character

Personal attributes of the true Christian

What determines a person's character?

Love of God and love of virtue are closely united

Intellectual disparities among people and the power in good habits

Tools of Satan. Tactics and Tricks used by the Devil.

The Natural Way -- The Unnatural Way

Wisdom, Reason and Virtue are closely related

Knowledge is one thing, wisdom is another

My views on Christianity in America

The most important thing in life is understanding

We are all examples --- for good or for bad

Television --- spiritual poison

The Prime Mover that decides "What We Are"

Where do our outlooks, attitudes and values come from?

Sin is serious business. The punishment for it is real. Hell is real.

Self-imposed discipline and regimentation

Achieving happiness in life --- a matter of the right strategies

Self-control, self-restraint, self-discipline basic to so much in life

[ Home ] [ Up ] [ Info ] [ Mail ]