Graphics Reference
In-Depth Information
Chapter 22
Curves
In this chapter and the next, we turn very briefly to the topics of splines and sub-
division, which are closely related. Both are used in
geometric modeling
(repre-
senting geometric shapes of the sort that we want to animate and render). Splines
are also used in image processing, animation, data fitting, and a host of other
applications. The web materials for these chapters provide a far more thorough
treatment of splines. In this chapter, we provide only the briefest outline of some
of the most common splines and subdivision curves; in the next, we will discuss
surfaces.
We begin with two widely used ways to specify a curve. You can think of these
as analogous to two ways to specify a line segment: You could specify the end-
points,
P
and
Q
, or one endpoint,
P
, and a vector
v
to the other endpoint (which is
therefore
Q
=
P
+
v
). Each form of specification has its uses, and both specify the
same geometric entity. Both are instances of the Coordinate-System/Basis princi-
ple: By choosing the correct basis in which to work (in this case a basis for the
vector space of cubic curves in the plane or space), we make our work simpler.
Curves: The Hermite Curve
Imagine that you're animating a car that's driving up the
y
-axis with velocity
03
T
, and arrives at the point
(
0, 4
)
at time
t
=
0. You need to animate its
595