• DOI:10.1147/RD.345.0736
• Corpus ID: 15349754

Pythagorean hodographs

@inproceedings{Farouki1990PythagoreanH,
  title={Pythagorean hodographs},
  author={Rida T. Farouki and Takis Sakkalis},
  year={1990}
}

• R. Farouki, T. Sakkalis
• Published 1 September 1990
• Mathematics

The hdograph of a plane parametric curve r(t) = (x(t), y(t)f 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 a(t) such that x t 2 (t) + y " (t) " Pythagorean triple. " Although Pythagorean-hodograph curves have fewer degrees of freedom than general polynomial curves of the same degree, they exhibit remarkably 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-Bezier forms, and survey the useful attributes it entails in various applications. ~ ' (t) , i.e., (x' (t), y ' (t) , a (t)) form a 1. Introduction The representation of curves and surfaces in a form amenable to efficient, systematic computation is a basic issue in computer-aided design. Those representations Topyright 1990 by International Business Machines Corporation. Copying in printed form for private use is permitted without payment of royalty provided that (1) each reproduction is done without alteration and (2) the Joz~rnal reference and IBM copyright notice are included on the first page. The title and abstract, but no other portions, of this paper may be copied or distributed royalty free without further permission by computer-based and other information-service systems. Permission to republish any other 736 portion of this paper must be obtained from the Editor. that have won widespread acceptance in practical use are almost exclusively parametric formulations, based on (piecewise) polynomial functions (see [ 11 and references therein). Plane curve segments, for example, are usually defined in a form equivalent to " x(t) = aktk, k=O n y (t) = 1 bktk for t E [0, 11. k=O Such segments may be pieced together with various orders of continuity to form spline curves for smooth data interpolation; they are easily rendered by uniformly incrementing t and evaluating the polynomials (1); and algorithmic procedures are available for computing their intersections (see [ 2 ]). An immediate shortcoming of the form (1)-its inability to accommodate conic loci other than the parabola [3]-may be remedied by allowing the rational form r(t) = (X(t)/W(t), Y(t)/ W (t)) , where X(t), Y(t), and W(t) are polynomials. …