平面代数曲线的PH-C曲线逼近

PH-C curve approximation of planar algebraic curves

代数曲线的近似参数化问题是计算机辅助几何设计与图形学领域的一个重要问题.由于PH-C曲线综合了Bézier曲线,PH曲线以及C曲线的许多优良性质,从而用PH-C曲线逼近代数曲线就显得十分必要.首先根据曲线的凹凸区间和单调区间对代数曲线进行合理分割,然后根据曲线段两端点的切线确定曲线段的三角形凸包,进一步根据此三角形凸包确定3次PH-C曲线的控制多边形,这样得到的PH-C逼近曲线保持了原代数曲线的一些重要几何性质,如单调性、凹凸性和G1连续性,并且通过算法的递归调用,可以将逼近误差控制在给定的范围之内.数值实验表明,该算法提供了平面代数曲线近似参数化的一条有效途径.

Approximate parameterization of algebraic curve is an important topic in computer aided geometric design and graphics.PH-C curve inherits all the good quality from Bézier curve,PH curve and C curve,therefore PH-C curve approximation of algebraic curve is necessary.The algebraic curve is segmented according to convexity and monotonicity,the control polygon is constructed based on angles between two tangent lines and the line connecting two end points.A detailed algorithm is proposed to approximate algebraic curve with piecewise degree 3 PH-C curve.The approximate PH-C curve keeps some important geometric features of the original algebraic curve such as convexity、monotonicity and G1 continuity.The approximation error can be controlled by means of recursively use of the algorithm.Numerical experiments show that the algorithm provided an efficient approach to approximate parameterization of algebraic curve.