7.2 The de Casteljau algorithm

In projective space, b( t) may be evaluated by a projective de Casteljau algorithm:

(7.7)

with ( t) = b _i. Then ( t) = b( t) is the point with parameter value t. Fig. 7.6 illustrates the cubic case n = 3. ^[4] This de Casteljau algorithm is also defined in ; there it yields polynomial B zier curves for more details, see [57]. If we introduce

Figure 7.6: The projective de Casteljau algorithm a point on a curve is constructed by repeated linear interpolation.

we observe that

Just as we did to prove (4.6), we can define

(7.8)

and obtain that

(7.9)

As a consequence,

(7.10)

This shows that the projective de Casteljau algorithm (7.7) is projectively invariant: the point ( t) can be found by intersections and cross ratio constructions only. The points q _i are, of course, the projective pre-images of the weight points q _i.

We could also project every step of the projective de Casteljau algorithm into and obtain the rational de Casteljau algorithm:

(7.11)

with ( t) = b _i and . Then ( t) = b( t) is the point with parameter value t. Here, care must be taken in the case of t being outside the interval [0, 1], as some of the intermediate values may be vectors! ^[5] Example 7.1 illustrates this: the...