Graphics Reference
In-Depth Information
Chapter 22
Splines and Subdivision
Curves
22.1 Introduction
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.
22.2 Basic Polynomial Curves
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.
22.3 Fitting a Curve Segment between Two
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
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