Optimal multi-degree reduction of Bézier curves with geometric constraints

In this paper we present a novel algorithm for the multi-degree reduction of Bezier curves with geometric constraints. Based on the given constraints, we construct an objective function which is abstracted from the approximation error in L"2-norm. Two types of geometric constraints are tackled. With the constraints of G^2-continuity at one endpoint and G^1-continuity (or C^r-continuity) at the other endpoint, we derive the optimal degree-reduced curves in explicit form. With the constraints of G^2-continuity at two endpoints, the problem of degree reduction is equivalent to minimizing a bivariate polynomial function of degree 4. Compared with the traditional methods, we derive the optimal degree-reduced curves more effectively. Finally, evaluation results demonstrate the effectiveness of our method.