4.1 B zier Curves

A B zier curve of degree n is defined by

where the B ⁿ _i(t) are Bernstein polynomials

The binomial coefficients are defined as

In the cubic case, this is identical to (3.4). The Bernstein polynomials of degree four are shown in Figure 4.2. The control polygons in the figure are explained in a more thorough discussion of Bernstein polynomials in Section 4.9.

Figure 4.2: The Bernstein polynomials of degree four plotted over [0, 1].

Several examples of higher degree B zier curves are shown in Figure 4.3. These examples show how a user might influence the shape of a B zier curve by adding more control points or moving control points.

Figure 4.3: A sequence of B zier curves.

For general degrees, there are essentially no new properties to report; everything carries over from the cubic case. Be sure to review the properties listed in Section 3.2. Let's revisit the major topics from Chapter 3.

4.2 Derivatives Revisited

For the derivative, we have

where ? b _i = b _i+1 ? b _i. This is again a B zier curve; its degree is n ? 1 and its coefficients are vectors.

B zier curves may be differentiated more than once. The k ^th derivative at parameter value t is given by

where ? ^k is the forward difference...