TABLE OF CONTENTS Chapter 5 explored the types of curves that are generated by a direct generalization of the constructive process first developed for conics in Example 3.16. The resulting curves are the B zier curves. Positive features of the B zier form are the variation diminishing property, the convex hull property, and the straightforward constructive algorithm. These properties allow us to think of the curve as a smoothed version of the control polygon. One positive feature of the B zier form, that it results in a single polynomial curve, can also be a negative feature since
when the ordered point set defining the curve gets large, the degree of the curve is high, and
all polynomial bases, including the Bernstein basis, must be nonzero over the domain except at a finite number of points. Thus, changing the coefficient of any basis function modifies the whole curve. For a B zier curve this means that moving any one control point changes the whole curve. The global effect is lessened because of the particular features of the Bernstein basis, but it is there.
In this chapter we seek to examine the constraints on overcoming the above negative features while retaining positive features of the B zier curve design method. That is, we are seeking to retain the characteristics that allow the curve to be considered as a smoothed version of the control polygon. We investigate developing a representation that is piecewise polynomial and allows specification of degree of smoothness across the joins of the polynomial pieces. The techniques...
