Graphics Reference
In-Depth Information
tend to detect local imperfections whereas reflection lines catch more global prob-
lems. A related approach is the method of
isophotes
([Pösc84], [HutH96]) that ana-
lyzes lines of equal light intensity. There are, as one can see, a number of special curves
on a surface that can be used to analyze it instead of using the surface curvature
directly. Theisel and Farin ([TheF97]) show how one can determine the curvature of
contour lines, lines of curvature, asymptotic curves, reflection lines, and isophotes
without computing the curves directly. For more on surface interrogation algorithms
see [Hage92] or [FolR93]. Rather than having the designer use surface analysis tools
to fair a surface him/herself, Rando and Roulier ([RanR91]) describe a system that
fairs parametric surfaces automatically. Moreton and Séquin ([MorS92]) describe a
way to design smoothly shaped surfaces of any genus.
The problem of fairing discrete surfaces has also been studied. See [SuLi89]. A
discussion on the fairing of subdivision surfaces can be found in [ZorS99]. One moti-
vating observation is that obtaining a fair surface may be more important than obtain-
ing a smooth one. If one has a polygonal mesh, one has more freedom to make
changes. Discrete analogs of curvature are available. One can always get a smooth
surface at the end if that is needed, an interpolatory B-spline surface in fact.
Finally, if problems with a surface's shape are detected, they need to be corrected.
In the case of tensor product surfaces one can try some curve-smoothing methods on
the curves in the two coordinate directions. For much more about surface fairing see
[HosL93]. Section 15.2 has more about fairing and curvature.
12.17
Recursive Subdivision Surfaces
This section is about polygonal surfaces and ways to turn them into smoother looking
objects. There are basically two approaches. One (see, for example, [Pete95]) is to
define parametric patches for each facet and ensure that they all meet in a smooth
manner. The other uses a recursive subdivision process that smooths the corners and
edges to get a closer and closer polygonal approximation of a smooth surface. It is
these recursive subdivision surfaces that we want to discuss in this section. There are
two ways of looking at such surfaces. From one point of view, we can consider the
original polygonal surface as having been a rough outline of some specific smooth
one and that reconstruction of this smooth surface is our goal. One reason that one
may not have had or even care about an actual parameterization for the smooth
surface is that surfaces that have reasonable parameterizations with the usual rec-
tangular or triangular domains are somewhat limited in their shape or topology unless
one is willing to put up with what may be unpleasant singularities. Even a space as
simple as a sphere cannot be parameterized without singularities unless one uses
several patches to cover it. The other point of view is that recursive subdivision sur-
faces are a wholly new class of surfaces that are interesting in their own right inde-
pendent of any associated smooth surface.
A number of polygonal surface subdivision algorithms are known. An excellent
overview of the subject and its applications can be found in [ZorS99]. See also
[CavM89] and [Sabi90]. One large class of such algorithms can be classified by
whether they use a “corner cutting” or vertex insertion approach. In the latter case
one can distinguish further based on whether they generate quadrilateral or
