Some improvements on optimal multi-degree reduction of Bézier curves with geometric constraints

Recently, Zhou et al. (2014) presented an efficient method for optimal multi-degree reduction of Bézier curves with geometric constraints. We notice that their method is unable to always preserve the original tangent directions at the endpoints, which is a desirable shape preserving property in geometric modeling and related applications. In order to satisfy this requirement, we impose a feasible region for the two parameters relating to the G1 constraint and convert degree reduction with the feasible region into a constrained minimization problem. The projected Newton method employed for solutions exhibits faster convergence. Our method can produce the desired results which, however, cannot be achieved by the previous method in a few cases.