Pythagorean-hodograph space curves

Abstract

We investigate the properties of polynomial space curves r(t)={x(t), y(t), z(t)} whose hodographs (derivatives) satisfy the Pythagorean condition x′2(t)+y′2(t)+z′2(t)≡σ2(t) for some real polynomial σ(t). The algebraic structure of the complete set of regular Pythagorean-hodograph curves in ℝ3 is inherently more complicated than that of the corresponding set in ℝ2. We derive a characterization for all cubic Pythagoreanhodograph space curves, in terms of constraints on the Bézier control polygon, and show that such curves correspond geometrically to a family of non-circular helices. Pythagorean-hodograph space curves of higher degree exhibit greater shape flexibility (the quintics, for example, satisfy the general first-order Hermite interpolation problem in ℝ3), but they have no "simple" all-encompassing characterization. We focus on a subset of these higher-order curves that admits a straightforward constructive representation. As distinct from polynomial space curves in general, Pythagorean-hodograph space curves have the following attractive attributes: (i) the arc length of any segment can be determined exactly without numerical quadrature; and (ii) the canal surfaces based on such curves as spines have precise rational parameterizations. © 1994 J.C. Baltzer AG, Science Publishers.