Pythagorean-Hodograph Quintic 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, but such a curve cannot inflect. Quintic PH curves are needed to get an inflexion point. The degrees of freedom of such a curve allow the choice of four of the six control polygon points. The PH property determines the two remaining points.
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 . A PH curve needs to have degree five to have an inflexion point. These quintic curves are written in BP form (see Related Links), that is, represented by their control polygons .
Denote the distance between and by and the angle by . A quintic curve is a PH curve if and only if , , and . These equations determine the last two control points based on the previous four.
 R. T. Farouki, Pythagorean-Hodograph Curves: Algebra and Geometry Inseparable, Berlin: Springer, 2008.