ABSTRACT
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 Bézier curves. Positive features of the Bézier 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 Bézier 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 Bézier 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.
