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Pythagorean-Hodograph Curves: Algebra and Geometry Inseparable
Pythagorean-Hodograph Curves: Algebra and Geometry Inseparable
Date: 11 November 2010, 16:02

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Pythagorean–hodograph curves are characterized by the special property that their “parametric speed”—i.e., the derivative of the arc length with respect to the curve parameter—is a polynomial (or rational) function of the parameter. This distinctive attribute, achieved by a priori construction of the hodograph (derivative) components of polynomial or rational curves in Rn as elements of Pythagorean (n+1)–tuples, endows the Pythagorean–hodograph (PH) curves with many computationally attractive features. For example, it is possible to compute their arc lengths, bending energies, and offset (parallel) curves in an essentially exact manner, without recourse to approximations; and they are exceptionally well–suited to problems of real–time motion control and spatial path planning based on the use of rotation–minimizing frames.
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