Pythagorean-Hodograph (PH) Cubic Curves
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.
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 ).
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.
 R. T. Farouki, Pythagorean-Hodograph Curves: Algebra and Geometry inseparable, Berlin: Springer, 2008.