Pythagorean-Hodograph (PH) Cubic Curves

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A polynomial curve is a Pythagorean-Hodograph (PH) curve if is the square of another polynomial. The lowest-degree curves satisfying this condition are PH-cubics, here presented in Bézier form. The degrees of freedom of such a curve let you choose three of the four control polygon points. The remaining point is then determined.

Contributed by: Isabelle Cattiaux-Huillard and Gudrun Albrecht (March 2014)
Open content licensed under CC BY-NC-SA



Consider a polynomial parametric curve . By definition, its hodograph is its derivative . It is said to be Pythagorean if there is another polynomial such that . The curve is then said to have a Pythagorean hodograph, or, for short, to be a PH curve. it has the remarkable properties of having polynomial speed and permitting offset curves with rational parametrizations.

The lowest-degree curve allowing this property is three. Therefore this Demonstration shows cubic curves written in Bézier form, that is, represented by their control polygons (see [1]).

Denoting by the distance between and and by the angle , a cubic curve is PH if and only if and . This result allows the free choice of three of the control points; the fourth one is then determined.


[1] R. T. Farouki, Pythagorean-Hodograph Curves: Algebra and Geometry inseparable, Berlin: Springer, 2008.

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