Curves in Computer Graphics - Computer Graphics and Geometric Modeling - page 449

basically starting the curve design from scratch again. The fact is that the user may
already have a reasonable shape and may only want to fine tune it. What is needed
therefore is that one wants to first add control points near the point of interest in such
a way that the larger collection still describes the original curve and then let the user
continue with more detailed modifications. Fortunately, this is possible. One can
increase the number of control points for a Bézier curve and still end up with the
same curve. We shall not describe this method, called “degree elevation,” here. See
[Fari97]. Instead, we describe a curve subdivision method.
Given that [0,1] is the domain for the curve p(u), we shall again divide the domain
[0,1] at a value c, 0 £ c £ 1. The set p([0,c]) is actually the path of a Bézier curve q(v)
defined on [0,1] with control points q 0 = p 0 , q 1 ,..., q n = p(c), so that

q u
c

Ê
Ë

ˆ
¯

() =

Π[] pu , uc 0 , .

The problem is to find the control points q i . See Figure 11.33. Using the notation of
the de Casteljau algorithm in Section 11.5.2:

i

()

qp = c (11.128)

i

0

This shows that the de Casteljau algorithm can be used to find the q i . Furthermore,
because of the symmetry property of Bézier curves, the control points r 0 = p(c),
r 1 ,..., r n = p n of the Bézier curve

() =+- ( ( ) ) Π[]

rv pc v 1 c , v 0 1 , ,

for the set p([c,1]) are

i

() rp = c . (11.129)

i

ni -

The points q i and r j are the extra control points we were looking for.
The subdivision problem for B-spline curves has a solution similar to that of the
Bézier problem. We already gave two algorithms in Section 11.5.2 for adding either
a single knot (Theorem 11.5.2.13) or multiple knots (Theorem 11.5.2.14).

11.9 Composition of Curves and Geometric Continuity

From an abstract point of view, when one talks about curves one usually has one-
dimensional subsets of R m in mind and the parameterizations of these subsets are
incidental. One is interested in properties of these sets . The functions that parame-
terize them are usually just intermediary concepts. In practice however, parameteri-
zations do play a role. A given curve may be defined by means of several parametric
curves, each of which traces out only part of the whole curve. Questions arise as to
how the parametric curves meet. This section takes a brief look at some answers to
such questions.