Natural Extension of Rational Bézier Curves

The natural extension of a freeform curve has the advantageous properties of preserving both the number of control points and the parameterization. To extend a rational Bézier curve to a specified length, a method to calculate the natural extension of a given curve through the reparameterization technology is proposed. First, the curve parameter corresponding to the extension length is calculated by using the iteration method from the given curve and extension length. Then, the explicit expression of the extended curve is derived by using the reparameterization technology, which remains identical to that of the original curve. Finally, the weight factors and control points of the extended curve are computed respectively by using the de Casteljau algorithm according to Theorem 1. Experimental results on traditional arcs, cylinders and rational curves and surfaces implemented in C++ are given to show the performance of our algorithm for curve extension applications, which demonstrates that the number of control points in the extended curve matches that of the given curve. Geometrically, the extended curve preserves the geometric shape of the given curve, with the parameterization of the given part being linearly scaled.