二阶几何连续的闭合全凸曲线的构建

Construction of Closed, Global Convexity and G^2 Continuity Curve

针对现有保凸曲线插值算法不能解决过平面凸包点集构建闭合全凸光滑曲线的实际应用问题,提出一种二阶几何连续的闭合全凸曲线的插值算法.该算法以一个平面凸包点集为插值点,以相邻的2个凸包点作为1条3次Bézier曲线的第1个与第4个控制点,根据相邻3次Bézier曲线间的二阶几何连续性条件求解每条3次Bézier曲线的第2个与第3个控制点;然后从理论上证明了曲线的闭合性、全凸性及二阶几何连续性,并提出一种简易有效的曲线构建算法.实验结果表明,该插值曲线具备明确的物理学意义上的解释;将该算法应用于模拟卷尺测量轨迹以提取树干直径的实际场景中,进一步验证了其精确性与实用性.

existing methods of curve interpolation cannot solve practical application problems of constructing a closed smooth curve with global convexity for planar convex hull point set.For this purpose,a curve interpolation algorithm for constructing a closed G2continuity curve with global convexity is proposed.A planar convex hull point set was used as interpolating points.The two adjacent convex hull points were used as the first and the fourth control points of a cubic Bézier curve,and the second and the third control points were resolved by the relationship of geometric continuity between the two adjacent Bézier curves.The closed,G2continuity and global convexity properties of the constructed curve were proved theoretically.A simple and effective curve constructing algorithm was presented.The experiment showed that the constructed curve has an explicit physical explanation.The practical application of simulating the measurement path of tape to retrieve stem diameter by the constructed curves verifies the accuracy and practicability of the proposed curve interpolation algorithm.