102 Vectors and geometry in three dimensions
particular coordinate system is better than others and that there exist
formulas relating the coordinates in two different coordinate systems.
It is sometimes said that fundamental problems in analytic geometry
are not adequately covered in textbooks that combine the study of
analytic geometry and calculus. Much more analytic geometry will
appear later in this textbook. Meanwhile, we consider a fundamental
problem in analytic geometry that is sometimes ignored in elementary
geometry books. Suppose we say, with reference to some rectangular
x, y, z coordinate system, that a set S in E3 is a quadric surface if it is
the set whose points P(x,y,z) satisfy an equation of the form
(2.68) 4x2+Bye+Cz2-}-Dxy+Exz+Fyz+Gx+Hy+Iz+J = 0,
where the coefficients A, B, C, D, E, F are not all zero. Our big question
is the following. Can it happen that Miss White chooses a particular
x, y, z coordinate system and finds that a particular set S* is a quadric
surface because there do exist coefficients 11, B, , F not all 0 such
that S* is the set of points P(x,y,z) for which 11x2 + By2 + ... = 0,
while, at the same time, Mr. Black chooses another x, y, z coordinate
system and finds that the same set S* is not a quadric surface because
for his system the required coefficients do not exist? If the answer is
affirmative, then the above definition of quadric surface and the above
set S* should be placed in the museum of the SPC (Society for the Pro-
motion of Confusion). It can be shown that the answer is negative
and hence that the definition of quadric surface does make sense. To do
this, we let x', y', z' denote the coordinates of Mr. Black and substitute
the values of x, y, z from (2.67) into (2.68) to find what the equation of
S* will be in the coordinates of Mr. Black. The critical equation turns
out to be(2.681)whereA'x'2 + B'y'2 + C'z'2 + D'x'y' + E'x'z' + F'y'z'
+ G'x' + H'y' + I's' + J' = 0,
I' = flail + Bata + Ca13 + Dana,, + Eaua,3 + Fa,2a,3,
and formulas for the other coefficients can be written out. Proof that
the coefficients A', B', C', D', E', F' are not all zero can be based upon
the fact that substituting the expression for x', y', z' from (2.671) into
(2.681) must yield the original equation (2.68). If A', B', C', D', E', F'
were all zero, this substitution would show that A, B, C, D, E, F are all
zero, and this is not so. The principle involved is the following: As we
see from (2.671), a change from coordinates x, y, z to x', y', z' cannot
increase the degree of a polynomial in x, y, z. Moreover, the change
cannot decrease the degree because, as we see from (2.67), the change
from x', y', z' back to x, y, z cannot bring a polynomial of lower degree
back to the original polynomial.