Approximating Curves

It might seem like the easiest way to control a curve is to specify a set of points for it to interpolate. In practice, however, interpolation schemes often have undesirable properties because they have less continuity and offer no control of what happens between the points. Curve schemes that only approximate the points are often preferred. With an approximating scheme, the control points influence the shape of the curve, but do not specify it exactly. Although we give up the ability to directly specify points for the curve to pass through, we gain better behavior of the curve and local control. The two most important types of approximating curves in computer graphics are Bezier Curves and B-Spline Curves.

Page  7  (Bezier Curves) Page  8  (B-Splines)

Exercises

1. Basis Matrix for Cubic Bezier: The constraints for a segment of a Bezier cubic are:
   $$\begin{array}{rl} f(0) & = p_0\\ f'(0) & = \frac{1}{3} (p_1 - p_0)\\ f(1) & = p_3\\ f'(1) & = \frac{1}{3} (p_3 - p_2) \end{array}$$

   (1)
   Using the methods of Section Polynomial Pieces, derive the basis matrix and blending functions for a Bezier cubic segment.
2. Convert a Hermite Cubic to a Bezier: Given the four control points of a segment of a Hermit spline, compute the control points of an equivalent Bezier segment.
3. De Castijeau Algorithm: Use the De Castijeau algorithm to evaluate the position of the cubic Bezier curve with its control points at (0,0), (0,1), (1,1) and (1,0) for parameter values $u=.5$ and $u=.75.$ Drawing a sketch will help you do this.
4. Cox / de Boor Recurrence: Use the Cox / de Boor recurrence to derive Equation Equation (See below).