Pythagorean hodograph curve

In mathematics, a Pythagorean hodograph curve or PH curve is a curve defined by a polynomial parametric equation for which the speed (the derivative of arc length) also has a polynomial parametric equation. This allows the arc length itself to be determined by integrating the speed. Additionally, dividing by the speed gives a rational parameterization of the unit normal to the curve, and of the parallel curves to the given curves. For these reasons, researchers have investigated the use of Pythagorean hodograph curves as splines in geometric design.

Here, "hodograph" is another word for a derivative. They are called Pythagorean hodograph curves because their derivatives obey an equation analogous to the equation in the Pythagorean theorem.

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