Linear methods for G ¹, G ², and G ³ - Multi-degree reduction of Bézier curves

In this paper, linear methods to find the multi-degree reduction of Bézier curves with G ¹-, G ²-, and G ³-continuity at the end points of the curves are derived. This is a significant improvement over existing geometric continuity degree reduction methods. The general equations for G ²- and G ³-multi-degree reduction schemes are non-linear; we were able to simplify these non-linear equations to linear ones by requiring C ¹-continuity. Our linear solution is given in an explicit, non-iterative form, and thus has lower computational costs than existing methods which were either non-linear or iterative. Further, there are no other existing G ³-methods for multi-degree reduction. We give some examples and figures to demonstrate the efficiency, simplicity, and stability of our methods.