7.1 The Bernstein Form

Any n th degree polynomial curve b( t) in may be expressed in Bernstein form:

(7.1)

where the are the Bernstein polynomials

(7.2)

It is customary to add the following to this definition:

We note two important properties of Bernstein polynomials: they form a partition of unity:

(7.3)

and they satisfy the recursion

(7.4)

for more properties, see [57].

Figures 7.1, 7.2, and 7.3 give some examples of B zier curves in ; the 3D cases have a similar flavor. We show the complete curves in most practical applications, one would only be interested in the segments "inside" the control polygon, i.e., those points corresponding to t ? [0, 1] in (7.1).

Figure 7.1: B zier curves a degree four (quartic) example.

Figure 7.2: B zier curves another quartic example.

Figure 7.3: B zier curves a quintic example.

We may project (7.1) into affine three-space: we will then obtain a rational B zier curve b( t). This follows the same development as that of rational quadratics in Section 5.1, and yields

(7.5)

The w _i are now called weights. If they are all positive, the curve lies in the convex hull of the control polygon b ₀, ... , b _n. ^[2] This is called the convex hull property of B zier curves.

If one weight, w _k, say, is increased, the curve is "pulled" towards b _k.

We can make a more precise statement as...