Hermite interpolation with simple planar PH curves by speed reparametrization

We introduce a new method of solving C¹ Hermite interpolation problems, which makes it possible to use a wider range of PH curves with potentially better shapes. By characterizing PH curves by roots of their hodographs in the complex representation, we introduce PH curves of type K(t?c)²ⁿ⁺¹+d. Next, we introduce a speed reparametrization. Finally, we show that, for C¹ Hermite data, we can use PH curves of type K(t?c)²ⁿ⁺¹+d or strongly regular PH quintics satisfying the G¹ reduction of C¹ data, and use these curves to solve the original C¹ Hermite interpolation problem.