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15.1. Curves 341
reason, mathematicians typically are careful to distinguish between a curve and
its representations. In computer graphics we are often sloppy, since we usually
only refer to the representation, not the actual curve itself. So when someone says
“an implicit curve,” they are either referring to the curve that is represented by
some implicit function or to the implicit function that is one of the representations
of some curve. Such distinctions are not usually important, unless we need to
consider different representations of the same curve. We will consider different
curve representations in this chapter, so we will be more careful. When we use a
term like “polynomial curve,” we will mean the curve that can be represented by
the polynomial.
By the definition given at the beginning of the chapter, for something to be a
curve it must have a parametric representation. However, many curves have other
representations. For example, a circle in 2D with its center at the origin and radius
equal to 1 can be written in implicit form as
f(x, y)=x
2
+ y
2
− 1=0,
or in parametric form as
(x, y)=f(t)=(cost, sin t),t∈ [0, 2π).
The parametric form need not be the most convenient representation for a given
curve. In fact, it is possible to have curves with simple implicit or generative
representations for which it is difficult to find a parametric representation.
Different representations of curves have advantages and disadvantages. For
example, parametric curves are much easier to draw, because we can sample the
free parameter. Generally, parametric forms are the most commonly used in com-
puter graphics since they are easier to work with. Our focus will be on parametric
representations of curves.
15.1.1 Parameterizations and Re-Parameterizations
A parametric curve refers to the curve that is given by a specific parametric func-
tion over some particular interval. To be more precise, a parametric curve has a
given function that is a mapping from an interval of the parameters. It is often
convenient to have the parameter run over the unit interval from 0 to 1. When the
free parameter varies over the unit interval, we often denote the parameter as u.
If we view the parametric curve to be a line drawn with a pen, we can consider
u =0as the time when the pen is first set down on the paper and the unit of time
to be the amount of time it takes to draw the curve (u =1is the end of the curve).