Graphics Reference
In-Depth Information
Figure 11.1.
A curve with a differentiable parameterization.
such as the one shown in Figure 11.1, which have differentiable parameterizations
but also have a cusp. Section 11.10 discusses how one can detect that condition. The
same thing can happen in the case of surfaces. Smoothness of the function p does not
imply smoothness of its range X (and vice versa). At the heart of this is again the dis-
tinction between a C k
function and a C k
manifold, which is a set . We shall take a
closer look at this issue in Section 11.9.
11.1
Introduction to Curves and Surfaces
The first 10 chapters of this topic have described the basic ideas and algorithms cur-
rently used to render geometric objects. The main topics covered were the mathe-
matics for the graphics pipeline, clipping, drawing discrete lines, visible surface
determination, and shading. In short, we know pretty much all that we need to know
to render any linear polyhedra. Linear polyhedra are a too-limited domain, however,
even if we were to include the conics. The time has come to talk about general “curved”
objects. After all, most interesting objects are curved.
There are many aspects to the study of curves and surfaces. We shall touch on a
lot of them in the next two chapters because curves and surfaces are clearly central
to geometric modeling. However, the subject and the literature dealing with it are
especially large and we do not want to raise overly high expectations with respect to
the coverage. Readers who become interested in a really in-depth discussion of certain
topics may not find enough here, especially with regard to all the various choices and
some practical details related to the most efficient algorithms and implementations.
They should not be surprised to find themselves reaching for one of the references for
this depth. Given the breadth of our overview of geometric modeling, it was just not
possible to do more here. It is hoped, however, that we shall have at least conveyed
the essence of the forest of fundamental ideas if not of the trees. Specifically, at the
end of the day, we expect the reader to have learned the following:
(1) the ability to describe the main curves and surfaces that one encounters in
the real world of manufacturing and other areas of computer graphics,
(2) a basic understanding of what makes these objects tick,
(3) efficient and robust algorithms to compute some of their most important
properties,
(4) ways to make it easier to manipulate these objects, because it is not just math-
ematicians that use the objects, and
(5) an appreciation of the richness of the subject.
We begin with some general comments. The first question that needs to be
answered is how curved objects should be represented. A polygon could be repre-
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