Representing conics by low degree rational DP curves

A DP curve is a new kind of parametric curve defined by Delgado and Pena （2003）; Jt has very good properties when used in both geometry and algebra, i.e., it is shape preserving and has a linear time complexity for evaluation. It overcomes the disadvantage of some generalized Ball curves that are fast for evaluation but cannot preserve shape, and the disadvantage of the B6zier curve that is shape preserving but slow for evaluation. It also has potential applications in computer-aided design and manufacturing （CAD/CAM） systems. As conic section is often used in shape design, this paper deduces the necessary and suffi- cient conditions for rational cubic or quartic DP representation of conics to expand the application area of DP curves. The main idea is based on the transformation relationship between low degree DP basis and Bemstein basis, and the representation tbeory of conics in rational low degree B6zier form. The results can identify whether a rational low degree DP curve is a conic section and also express a given conic section in rational low degree DP form, i.e., give positions of the control points and values of the weights of rational cubic or quartic DP conics. Finally, several numerical examples are presented to validate the effectiveness of the method.