Graphics Reference
In-Depth Information
Figure 14.4.
Distance from a curve to a
surface.
Figure 14.5.
Distance between two
surfaces.
(
)
¥¥
(
)
=-
(
)
¥=
pq
-
p
p
pq n
0
0
u
v
p
(
)
¥¥
(
)
=-
(
)
¥=
pq
-
q
q
pq n
(14.7)
u
v
q
See Figure 14.5. Again we need to check distances for the boundaries, namely, the dis-
tance of p|∂([a
1
,b
1
] ¥ [c
1
,d
1
]) to q(u,v) and the distance of q|∂([a
2
,b
2
] ¥ [c
2
,d
2
]) to p(u,v).
As an example of how one can improve the efficiency of these algorithms in special
cases see [KimK03]. Kim describes algorithms for computing the distance between a
canal surface and a plane, sphere, cylinder, cone, and torus.
Finally, for objects defined implicitly by equations f(
p
) = 0 and g(
q
) = 0 we must
find the simultaneous solutions to these two equations.
14.3
Polygonizing Curves and Surfaces
In previous chapters we have already seen a number of algorithms that applied to
smooth objects but which used linear approximations to these objects to accomplish
