Graphics Reference
In-Depth Information
Step 4.
The hypersurface defined by equation (13.28) can be parameterized (with
homogeneous coordinates) by
(
)
=
xrst
,,
,,
,,
r
(
)
=
yrst
s
(
)
=
zrst
t
(
)
Hrst
Hrst
,,
,,
2
1
(
)
=-
(13.29)
wrst
,,
.
(
)
Substituting these functions into the formula for G, we get a homogeneous plane
curve. Dehomogenizing this curve gives us an affine plane curve that is now solved.
13.5.5.1 Example.
Consider the ellipsoid
S
1
centered at the origin and the sphere
S
2
centered at (1,0,0) defined by
(
)
=++-=
2
2
2
fxyz
,,
4
x
y
z
1
0
(13.30)
and
2
(
)
=-
(
)
22
2
22
gxyz
,,
x
1
++-=-++=
y
z
1
x
2
x y
z
0
,
(13.31)
respectively. Figure 13.19 shows the x-z plane cross-section of the two surfaces. We
show how to use Steps 1- 4 above to map the intersection of
S
1
and
S
2
to a planar
curve. (Of course, because the equations are so simple, we could have done this
directly without following any fancy steps, but this is beside the point.)
Solution.
Step 1 produces
(
)
=++-
2
2
2
2
Fxyzw
,,,
,,,
4
x
y
z
w
(
)
=-
2
2
2
Gxyzw
x
2
xw y
++
z
.
Step 2, based on the variable x, leads to
Figure 13.19.
The ellipsoids of Example
13.5.5.1.
