Graphics Reference
In-Depth Information
z
2
=
0
(3.43a)
(3.43b)
(3.43c)
2
2
xz
±=
0
2
2
2
xyz
+±=
0
Proof.
By Theorem 1.9.11, any symmetric matrix, the matrix A in (3.41) in particu-
lar, is congruent to a diagonal matrix with ±1 or 0 along the diagonal. This clearly
implies the result since A is not the zero matrix. Note that since we are working in
projective space here the change of coordinates transformations that produce equa-
tions (3.43) are
projective
and not affine transformation in general.
Translating things back to Euclidean space, it is easy to see that equations (3.43a)
and (3.43b) correspond to cases where the solution set to (3.35) is either empty or
consists of lines. Equations (3.43c) is the case where A is nonsingular.
Definition.
The affine conic defined by equation (3.35) or the projective conic
defined by equation (3.40) is said to be
nondegenerate
if the matrix A in (3.41) is non-
singular; otherwise it is said to be
degenerate
.
Note.
The definition of a nondegenerate conic has the advantage of simplicity but
has a perhaps undesirable aspect to it, at least at first glance. If A is nonsingular, then
one possibility for equation (3.43c) is
(
)
2
2
2
2
2
xyz
++=
0
rxy
++=
1
0
using Cartesian coordinates .
This equation has no
real
nonzero solutions. Therefore, a “nondegenerate” conic
could be the empty set. For that reason, some authors add the condition that a conic
be nonempty before calling it nondegenerate. On the other hand, we would get a non-
empty set if we were to allow complex numbers and we were talking about conics in
the complex plane. See Section 10.2.
We have just seen that matrix A in (3.41) determines one important invariant for
conics, but there is another. Consider the quadratic form that is the homogeneous
part of equation (3.35), namely,
(
)
=+ +
2
2
q
x y
,
ax
2
hxy
by
.
(3.44)
2
Let
ah
hb
=
Ê
Ë
ˆ
¯
B
(3.45)
be the matrix associated to q
2
. By Theorem 1.9.10 there is a change of basis that will
diagonalize B, that is, in the new coordinate system q
2
will have the form
