This repository is about Indirect PH Curves, a way to draw them in Matlab and confront them with quintic curves with Pythagorean hodographs.
Planar Bézier curves that have rationally parameterized offsets can be classified into two classes. The first class is composed of curves that have Pythagorean hodographs (PH) and the second class is composed of curves that do not have PHs but can have rational PHs after reparameterization by a fractional quadratic transformation.
We have derived the geometric characteristics of a class of cubic Bézier curves, called indirect-PH curves, which are not PH curves but can have rational Pythagorean hodographs after a quadratic parameter transformation. The characteristics are given in terms of Bézier control polygon geometry. The offsets to this class of Bézier curves are generally degree 8 rational curves. The construction of G1 Hermite interpolation using such curves is described. A single cubic indirect-PH curve has just one more degree of freedom than a quadratic Bézier curve, but bi-indirect- PH curves possess enough flexibility for G1 Hermite interpolation and shapes containing an inflection point.
Xing-Jiang Lu, Jianmin Zheng, Yiyu Cai, Gang Zhao. Geometric characteristics of a class of cubic curves with rational offsets. Computer-Aided Design 2016; 70:36-45.
Work made in 2018.