7.5 Reparametrization

We have seen that an arc of a conic does not have a unique parametric representation: depending on the choice of the shoulder tangent, each weight w _i could be multiplied by the power c ⁱ of a nonzero constant c: this would not change the geometry of the curve, but it would change how the parameter traverses it. See Section 4.9 for details. The same principle also serves to reparametrize B zier curves, in projective space as well after being projected into affine space.

We can write a projective B zier curve in the following two ways:

which means that a rational B zier curve can be written as

Reparametrization of a rational B zier curve changes the rate in which the parameter t traverses the curve; Fig. 7.9 illustrates.

Figure 7.9: Reparametrization a rational B zier curve is sampled at 50 equal parameter intervals. Left, c = 1; right c = 1/3.

After a curve has been reparametrized, it will have new weight points . Just as in the conic case, the old and new weight points are related by a constant cross ratio relation:

(7.22)

The old and new parameters of the curve are related by the same rational linear, or Moebius, transformation (4.37) that was used for the conic case:

(7.23)

A graph of the rational linear function (7.23) is shown in Fig. 7.10. ^[7]

Figure 7.10: Reparametrization the relationship between old and new parameter depending on the constant c.

A rational B zier...