Curves in Computer Graphics - Computer Graphics and Geometric Modeling - page 374

(2) A mixture of (1) and smoothness conditions, such as conditions that the deriv-
atives of g and f agree at the x j .
(3) Orthogonality constraints:

( ) ∑=

f - g j 0 for all i .

i

(4) Variational constraints:

fg -= min fh -

hA Œ

(5) Intuitive shape constraints involving, for example, the curvature of the curve
or surface.

A common thread, alluded to earlier, that underlies much of the discussion of
parametric curves and surfaces in CAGD, is:

Although it may seem like we are discussing different parameterizations, we are often talking
about one single function throughout and it is not the case that we are describing different
functions. The only thing that changes is how we represent the parameterization —which
control points we choose, what knots there are, if any, etc. Terms such as “Bézier curve” or
“B-spline curve” simply refer to different ways of looking at the same function.

The reader will find it helpful to keep this in mind. The study of curves and surfaces
in the context of CAGD largely revolves around coming up with techniques for
letting the user control their shape in the manner that is most natural for achieving
one's current ends. Furthermore, it involves finding ways to switch between various
representations. We shall see that polynomials are the most popular functions
used for parameterizations. The reason is that they are relatively simple to compute.
Furthermore, they play the same role for functions as integers play for real
numbers.
We are almost ready to start our study of curves, but first some terminology.
Consider a parametric curve

[ ] Æ

m

() = ( () () () )

pab : , R , pu pupu , ,..., p u .

1 2 m

Note that the component functions p i of p are just ordinary real-valued functions of
a real variable.

Definition. If all the p i have a certain property, then we shall say that p has that
property. For example, if all the p i are polynomials or splines (a term that will
be defined shortly), then we say that p is a polynomial or spline curve , respectively.
If all the p i are linear, quadratic, or cubic polynomials, then we say that p is a linear ,
quadratic , or cubic curve , respectively.

Our plan for this chapter is to start off with some simple examples of curves and
their properties that require no new knowledge past calculus. In particular, Sections