Construction of PH splines based on H-Bézier curves

There is considerable interest in the properties of PH curves in geometric modeling and CAGD because PH curves can be computed at speeds comparable to polynomial curves and used to calculate curve arc lengths and isometric lines. The purpose of this paper is to develop a general approximation of the H-Bézier curve based on PH curves. We call the resulting approximations PHH-Bézier curves for convenience. First, a necessary and sufficient condition for a cubic plane H-Bézier curve to be a PH curve is obtained. Second, based on the H-Bézier curve, the control polygon of the cubic PHH-Bézier curve is constructed from an orderly triangle. The cubic PHH-Bézier curve is then introduced and a new algorithm for their construction is proposed. According to geometric modeling, the error analysis between H-Bézier curve and PHH-Bézier curve is estimated. Finally, the proposed algorithm is verified experimentally. It is demonstrated that cubic PHH-Bézier curves can accurately approximate H-Bézier curves but that the selection of the middle two control points of the PHH-Bézier curve has a profound impact on the quality of the approximation. Error analyses demonstrate that when the middle two control points of the PHH-Bézier curve are mixed with the corresponding original control points, a good approximation is achieved. The new algorithm may thus have considerable potential for use in geometry modeling.