Consider a polynomial parametric curve
. By definition, its hodograph is its derivative
. The curve is called Pythagorean if there exists another polynomial
. 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 the two associated unit tangent vector directions, defined by the angles
, 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
do not give a solution.
 G. Jaklic, J. Kozak, M. Krajnc, V. Vitrih, and E. Zaga, "On Interpolation by Planar Cubic
Pythagorean-Hodograph Spline Curves," Mathematics of Computation
(269), 2010 pp. 305–326.
 R. T. Farouki, Pythagorean-Hodograph Curves: Algebra and Geometry Inseparable
, Berlin: Springer, 2008.