SMOOTH B-SPLINE MATCHING WITH ADDITIONAL CONSTRAINTS. PART 1: B-SPLINE AND ITS DERIVATIVES

Smooth conjugation of B-splines is one of the main functional operations in modern CAD-systems. An analytical method for smooth optimal conjugation of parametric curves represented as B-splines is proposed. Optimality is achieved by minimizing the metric of the difference between the original and conjugate splines. The neighborhood of the conjugation point is a parametric function (B-spline) belonging to the class of functions C ( p – 1), where p is the degree of the original splines. It is possible to supplement the optimization problem with pre-defined constraints that smoothly conjugate functions must satisfy. These constraints can be an exact match with some original spline and/or passing through a pre-specified point with pre-specified values of derivatives up to the (p – 1)-th order. Smoothly conjugate parametric functions are replaced without error by one resulting parametric function belonging to the class C ( p – 1). In this case, the number of degrees of freedom of the resulting parametric function is less than the sum of the degrees of freedom of the two original parametric functions. Conjugation can be applied to both plane and spatial curves.