Graphics Reference
In-Depth Information
where at least one of the six coefficients a, b, c, d, e, or f is nonzero. Such equations
define surfaces in general.
In trying to analyze the type of surface that equation (3.69) can give rise to, the
hard part is getting rid of the xy, xz, and yz cross terms. One can, like in the 2-
variable case, use a rotation to change the coordinate system to one in which the
equation has the form
2
2
2
ax
¢ ¢ +
by
¢ ¢ + ¢ ¢ + ¢ ¢+
cz
gx
hy
¢ ¢+ ¢ ¢+ ¢=
iz
j
0.
(3.70)
One could, for example, use the roll-pitch-yaw representation for a rotation about the
origin. This involves three unknowns. Substituting the rotated points into (3.69) and
then setting the coefficients of the cross terms to 0, would give three equations in three
unknowns which could be solved, but this is starting to get too complicated and messy.
The more elegant way to eliminate the cross terms in equation (3.69) or quadratic
equation in any number of variables is to use the theory of quadratic forms. Like in
the 2-variable case let
(
)
=+++++
2
2
2
q x y z
,,
ax
by
cz
dxy
exz
fyz
be the associated quadratic form. It follows again from Theorem 1.9.10 that q is
diagonalizable and there is a suitable coordinate system in which equation (3.69) has
the form (3.70). If any quadratic term is present in (3.70), then the corresponding
linear term, if there is one, can be eliminated by completing the square similar to the
way it was done in Section 3.6. The analysis depends on whether a¢, b¢, or c¢ are zero,
and if not, on whether they are positive or negative. The resulting cases are easy to
analyze and lead to the following theorem.
3.7.1. Theorem.
Any equation of the form (3.69) can be transformed via a rigid
motion into one of the following fourteen types of equations (equivalently, there is a
coordinate system, called the
natural coordinate system
for the quadric, with respect
to which equation (3.69) has the following form):
2
2
2
x
a
y
b
z
c
(1) Ellipsoid:
++=
1
2
2
2
2
2
2
x
a
y
b
z
c
(2) Hyperboloid of one sheet:
+-=
1
2
2
2
2
2
2
x
a
y
b
z
c
(3) Hyperboloid of two sheets:
--=
1
2
2
2
2
2
2
x
a
y
b
z
c
(4) Empty set:
-- -=
1
2
2
2
2
2
2
x
a
y
b
z
c
(5) Point:
++=
0
2
2
2
2
2
2
x
a
y
b
z
c
(6) Cone:
+-=
0
2
2
2
2
2
±=
x
a
y
b
(7) Elliptic or hyperbolic paraboloid:
cz where
,
c
π
0
2
