Optimal multi-degree reduction of Bézier curves with geometric constraints

• Optimal degree reduction of Bézier curves with various geometric constraints is presented.
• The degree-reduced curves are explicitly derived with some geometric constraints.
• With G2G2-continuity at two endpoints, our method is optimal and efficient.

In this paper we present a novel algorithm for the multi-degree reduction of Bézier curves with geometric constraints. Based on the given constraints, we construct an objective function which is abstracted from the approximation error in L2L2-norm. Two types of geometric constraints are tackled. With the constraints of G2G2-continuity at one endpoint and G1G1-continuity (or CrCr-continuity) at the other endpoint, we derive the optimal degree-reduced curves in explicit form. With the constraints of G2G2-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.