Graphics Reference
In-Depth Information
CHAPTER 12
Surfaces in Computer Graphics
Prerequisites: Basic calculus and vectors
Sections 8.1-5 in [AgoM05] (parameterization, surface, tangent vectors/
planes, manifolds)
Chapter 9 in [AgoM05] (curvature related properties, cyclides)
Terminology. See the important comments on terminology for curves at the begin-
ning of Chapter 11. Basically, the term “surface” by itself means a set and if it is pre-
ceded by an adjective, like in “parametric surface” or “Bézier surface” we mean a
function. By a surface patch we shall mean a parameterized surface whose para-
meterization is defined by a single formula.
12.1
Introduction
This chapter describes the main parametric and implicit surfaces that one encoun-
ters in computer graphics and geometric modeling. These are the surfaces that are
often referred to as “sculptured” or “free-form” surfaces to distinguish them from the
piecewise linear ones. By in large, the material is a natural extension of what was
developed for curves in the last chapter. For that reason it is important that the reader
understand the material in that chapter before starting this one. Many facts and algo-
rithms dealing with surfaces are easy if one understands their curve analogs.
Tangent planes for surfaces, in particular their normals, play an important role
just like the tangent lines did for curves. In the parametric case the planes will be
defined from certain partial derivatives associated to the parameterization. In the
implicit case, they will be defined from their equations.
There are many types of smooth surfaces and different ways to represent them.
Some representations involve interpolating data and others only approximate it. A
helpful way to organize the subject is by categorizing the general principles motivat-
ing the various representations. Using terminology not normally used by mathemati-
cians, but common in practice with engineers and craftsmen, three such principles
are lofting, superposition, and a tensor or Cartesian product representation. Ruled
Search WWH ::




Custom Search