Graphics Reference
In-Depth Information
12.16
Fairing Surfaces
Fairing surfaces is, like in the case of curves, a question of achieving a pleasing shape.
It is a much more complicated problem than fairing curves because the obvious
approach to reduce the problem to a one-dimensional one would involve the shape of
a surface along an infinite number of directions at every point. In any case, there are
again two sides to the problem. First, one needs some tools to detect any imper-
fections in the shape and, second, one needs to describe ways to correct these
imperfections.
Curvature is again key to the general detection process, but, just like the fact that
a simple number, such as the slope for real-valued functions, cannot capture the idea
of derivative for vector-valued functions, it is not easy trying to capture the idea of
curvature at a point on a surface with real-valued functions. Some basic functions
that have been used are
(1) Gauss curvature,
(2) mean curvature, and
(3) absolute curvature (Ωk 1 Ω+Ωk 2 Ω, where k 1 and k 2 are the principal normal
curvatures).
Determining imperfections in the fairness of a surface boils down to making sure that
plots of these curvature functions have appropriate shapes. In particular, places where
the sign of the curvature function changes too often are potential problem spots. A
problem with Gauss curvature is that it is zero for ruled surfaces and hence gives no
useful information in those cases.
Other specialized tools have been used. Hagen and Hahmann ([HagH95]) use sta-
bility concepts based on infinitesimal bendings to control the shape of a surface.
Séquin et al. ([SéCM95]) faired surfaces by minimizing functionals based on the arc-
length derivative of normal curvature, although this turned out to be very expensive
computationally. Other variational approaches are described by Sarraga ([Sarr98]).
Net fairing methods are described in [SuLi89] in case a surface is defined by a network
of curves.
Another method that has been used in designing pleasing car body shapes is based
on reflection lines ([Klas80]). Reflection lines are the patterns that one sees on the pol-
ished surface of a car caused by parallel lines of light from light sources. See Figure
12.28. The criterion for a nice shape is that these patterns look “nice.” Curvature plots
Figure 12.28.
Detecting surface
imperfections with
reflection lines.
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