Graphics Reference
In-Depth Information
CHAPTER 11
Curves in Computer Graphics
Prerequisites:
Basic calculus and vectors,
Section 5.2 in [AgoM05] (metric),
Sections 8.1-5 in [AgoM05] (parameterization, curve, tangent vector,
manifold),
Sections 9.1-4 in [AgoM05] (arc length, curvature)
Terminology.
The terms “curve” and “surface” get used in different ways. Sometimes
people use them to refer to sets, other times, to functions. Although it is usually clear
from the context which is meant, this ambiguity can lead to confusion because there
are times when that distinction plays a role. For that reason we want to use terms as
consistently as possible from the outset. An awareness of what is already known to
mathematicians is also important in graphics. A lack of such and perhaps a confu-
sion in the meaning of terms has given rise to some nonoptimal terminology in CAGD,
as, for example, in the case of “G
k
” curves and surfaces. Although our comments are
made for curves, similar comments hold for surfaces that are discussed in the next
chapter.
In this topic the term “curve” by itself always means a
set
. The potential ambi-
guity in terminology arises because such sets are usually described via functions and
one is tempted to use the same word “curve” for them. We shall allow that in specific
situations that we now describe, in order to be compatible with the way one actually
talks. The term “curve” preceded by appropriate adjectives will mean a
function
. In
particular, the term “parametric curve” encompasses all such functions and means a
continuous vector-valued
function
p : [a,b] Æ
R
m
defined on some interval [a,b]. In
this context, the expression “look at the curve” means “look at the set p([a,b]) that is
its range.” Other examples of expressions that refer to
functions
(parametric curves)
and
not
sets are “cubic curve,” “Bézier curve,” and “B-spline curve.” In addition, an
expression of the form “the curve p(u)” is shorthand for “the parametric curve p(u).”
Finally, the reader will recall from calculus that there is a difference between con-
tinuity and differentiability. Graphs of real-valued functions that are not differentiable
have “corners.” In geometric modeling there are times when we do not want our
curved objects to have corners. In analogy with the real-valued function case, a natural
reaction would be to assume that if we restrict our parameterizations to be differen-
tiable, then everything will be fine. Unfortunately, this is not so. There exist curves,
