TABLE OF CONTENTS In this chapter, we will see the main uses of parametric curves: describing geometric shapes using methods such as interpolation and approximation.
Suppose you are given four points p 0, p 1, p 2, p 3, and you wish to pass a curve through them, just like the situation shown in Sketch 33. There, the points are 2D, but they might as well be 3D. This is called interpolation.
We may choose among many kinds of curves; for right now, we'll use a cubic B zier curve. Every point on a B zier curve has a parameter value; in order to solve our problem, we have to assign a parameter value t i to every p i. A natural choice is to associate each p i with a parameter value t i = i/3. Many other suitable choices exist; [1] more on this topic in Section 5.4.
Now our interpolation problem becomes:
Given four point/parameter pairs ( p i, t i), find a cubic B zier curve x( t) such that
This simply states that we want the B zier curve to pass through the data points at the right parameter values.
The desired B zier curve will be of the form
Writing out all interpolation conditions (5.1) yields
These are four equations in the four unknowns b 0, , b 3. In order to...
