Curve interpolation using algebraic hyperbolic Pythagorean Hodograph curves of order four(四阶代数双曲毕达哥拉斯速端曲线插值方法) - 刘昊明(LIU Haoming); 徐帝纳(XU Dina); 方林聪(FANG Lincong); 李毓君(LI Yujun) (Zhejiang Daxue xuebao. Lixue ban)
This paper investigates geometric interpolation methods using algebraic hyperbolic Pythagorean Hodograph (AHPH) curves of order four, including G1 Hermite interpolation, C1 Hermite interpolation, and planar three-points interpolation. PH curves have been extended into the algebraic hyperbolic space as AHPH curves. Studying the interpolation methods using AHPH curves can enrich and improve geometric theories of PH curves. In the case of the G1 Hermite interpolation, within the real number domain, we construct a univariate quadratic real equation by employing the inner product of vectors and then solve it numerically. Regarding the C1 Hermite interpolation, we convert the problem of constructing AHPH curves into finding solutions of a complex univariate quadratic equation on a control point. The main idea is to adopt the complex representation of planar parametric curves and apply the necessary and sufficient conditions on control polygons of AHPH curves. For the interpolation problem with three planar data points, we assign the parameters to these points, thereby converting the interpolation conditions into initial conditions for searching the target curves within the AHPH curve family and constructing a quadratic complex equation to solve for the target curves. Consequently, for each of the abovementioned problems, there are no more than two AHPH curves satisfying the given interpolation conditions. We validate the effectiveness of the proposed algorithms with several numerical examples. Finally, by connecting AHPH curve segments, we construct several AHPH splines interpolating given planar point sets.研究了四阶代数双曲毕达哥拉斯速端（algebraic hyperbolic Pythagorean Hodograph，AHPH）曲线的几何插值方法，包括G1 Hermite插值、C1 Hermite插值以及平面三点插值。在代数双曲空间中，AHPH曲线是PH曲线的推广，研究AHPH曲线的插值方法可丰富和完善PH曲线的相关几何理论。对于G1 Hermite插值问题，在实数域下应用向量内积符号构建一元二次实方程并求解。对于C1 Hermite插值问题，应用平面参数曲线的复数表示方法和AHPH曲线控制多边形的充分必要条件，将曲线的构造问题转化为关于控制顶点的复一元二次方程进行求解。对于平面三点插值问题，通过插值点参数化将插值条件转化为在AHPH曲线族中搜索目标曲线的初值条件，进而构建一元二次复方程求解目标曲线。以上问题均不超过2条AHPH曲线满足插值条件。最后，通过数值实例，验证了本文方法的有效性。此外，应用本文方法可构造AHPH样条。