Consider a polynomial parametric curve

. By definition, its hodograph is its derivative

. The curve is called Pythagorean if there exists another polynomial

such that

. The curve is then said to have a Pythagorean hodograph or to be a PH curve. Therefore its speed

is also a polynomial function of

. The lowest degree allowing this property is three.

Hence we illustrate here how cubic curves, represented in Bézier form (see Related Link below) by their control polygons

, can be used for a

Hermite interpolation. Specifying the boundary points

and

and the two associated unit tangent vector directions, defined by the angles

and

, we determine the cubic interpolatory PH-curve by its control points

. In certain cases, such a curve cannot exist, because a cubic (PH) curve does not have an inflexion point, so some values of

and

do not give a solution.

[1] G. Jaklic, J. Kozak, M. Krajnc, V. Vitrih, and E. Zaga, "On Interpolation by Planar Cubic

Pythagorean-Hodograph Spline Curves,"

*Mathematics of Computation*,

**79**(269), 2010 pp. 305–326.

[2] R. T. Farouki,

* Pythagorean-Hodograph Curves: Algebra and Geometry Inseparable*, Berlin: Springer, 2008.