Construction of Cubic PH Curves Interpolating Three Points

This paper introduces a novel method to construct cubic PH curves by interpolating three planar points, which broadens their practical applications. Since these curves possess prominent geometric characteristics and algebraic properties, their necessary and sufficient conditions can be expressed by a complex system by employing complex representation of planar parametric curves. Following parametrization of given points, we get a quadratic complex equation from the complex system, which gives the control points of two cubic PH curves. We do experiments by constructing cubic PH curves for some given planar points. We compare different parametrization methods including uniform parametrization, chord length, and arc-length parametrization methods. In order to select the best solution, we further compute arc lengths, bending energies, and absolute rotation numbers of curves. Finally, as an application we use the proposed method to construct a cubic PH spline curve.