Pythagorean hodographs

Abstract

The hodograph of a plane parametric curve r(t) = {x(t), y(t)} is the locus described by the first parametric derivative r′ (t) = {x′ (t), y′ (t)} of that curve. A polynomial parametric curve is said to have a Pythagorean hodograph if there exists a polynomial σ(t) such that x′ <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> (t) + y′ <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> (t) ≡ σ <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> (t), i.e., (x′(t), y′(t), σ(t)) form a “Pythagorean triple.” Although Pythagorean-hodograph curves have fewer degrees of freedom than general polynomial curves of the same degree, they exhibit remarkable attractive properties for practical use. For example, their arc length is expressible as a polynomial function of the parameter, and their offsets are rational curves. We present a sufficient-and-necessary algebraic characterization of the Pythagorean-hodograph property, analyze its geometric implications in terms of Bernstein—Bézier forms, and survey the useful attributes it entails in various applications.