Hermite Interpolation with Pythagorean-Hodograph Cubic Curves

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A polynomial curve is a Pythagorean-hodograph curve if is the square of another polynomial. The lowest-degree curves satisfying this condition are PH-cubics. They are represented here in Bézier form. The degrees of freedom of such a curve allow using it to solve a partial Hermite interpolation problem: the boundary points and the tangent directions can be specified, but not the speeds at these points. Some situations have no solutions.

Contributed by: Isabelle Cattiaux-Huillard (June 2014)
Open content licensed under CC BY-NC-SA



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.

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