Graphics Reference
In-Depth Information
This finishes our discussion of contours, although much is also known for the
higher dimensional cases. As one reference for the contour surface problem we
mention [Rock90].
14.7
Skinning
Sections 13.4.3 and 14.6 dealt with what might be called finding level curves of a given
surface. This section turns the problem around.
Definition.
A
skinning surface
for a sequence of sets
C
0
,
C
1
,...,
C
k
is a surface
S
satisfying
(1)
S
interpolates the sets, and
(2) there is a continuous function f:
S
Æ [0,1] and real numbers 0 = t
0
< t
1
< ...
< t
k
= 1 so that
C
i
= f
-1
(t
i
).
The process of finding such a surface is called
skinning
. In practice, the sets
C
i
consist
of one or more curves, in which case they are sometimes called
skinning curves
.
In many cases, skinning is just a more recent term for lofting. Condition (2) of a
skinning surface is simply trying to capture the intent that the curves
C
i
should be
contours or level curves of the surface with respect to a function. Some skinning
algorithms may at times produce self-intersecting surfaces, but that would be an
undesirable “feature” of the algorithm. A common case for the curves is where they
are planar and correspond to parallel cross-sections of a surface. It is easy to imagine
though how complicated things might get if one is given an arbitrary collection of
curves.
Of course, like in all interpolation problems, one wants more from the skinning
surface
S
than that it interpolates the curves. Specifically, some properties that one
wants
S
to satisfy are:
(1) the shape of the surface should match the shape suggested by the curves and
it should not have any unnecessary wiggles.
(2) The surface should be smooth if all the curves are.
(3) The algorithm for getting
S
should be
affinely invariant
, meaning that if the
curves are moved by an affine transformation T, then the algorithm applied
to the new curves T(
C
i
) should produce T(
S
).
Skinning algorithms have been defined in two quite different contexts, a polygo-
nal and a smooth one. The polygonal case, which is often the harder one, is where
the curves are polygonal and we are looking for a faceted
S
. The smooth case is where
the curves are smooth parametric curves and one wants a smooth parameterization
for
S
. This section will only touch on a few aspects of the skinning problem. Two
papers that have numerous references to work on this subject are [MeSS92] and
[ParK96].
We consider the polygonal case of the skinning problem first. Meyers et al.
([MeSS92]) break this problem into four subproblems:
