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Journal for Geometry and Graphics 18 (2014), No. 1, 023--043
Copyright Heldermann Verlag 2014

Pythagorean Hodograph Curves: A Survey of Recent Advances

Jiri Kosinka
Computer Laboratory, University of Cambridge, 15 JJ Thomson Avenue, Cambridge CB3 0FD, England

Miroslav Lávicka
Dept. of Mathematics, NTIS -- Faculty of Applied Sciences, University of West Bohemia, Univerzitní 8, 301 00 Plzen, Czech Republic

Pythagorean hodograph (PH) curves, defined by Farouki and Sakkalis in 1990, and their applications in offsetting, CNC machining, motion planning etc., have become an active research area in recent years. These curves (and other objects with a Pythagorean property) have fascinated many researchers and one can find hundreds of papers devoted to this beautiful topic. An early survey on PH curves was done by Farouki in Section 17 of the Handbook of Computer Aided Geometric Design 2002. In 2008, a special issue of the journal CAGD was devoted to "Pythagorean-hodograph curves and related topics". A unified introduction to existing material on shapes with the Pythagorean property and also new research starting points were described in the book by Farouki in the same year. This book can also serve as a detailed survey of the literature concerning PH curves and related topics. However, development never stops and recent years have seen further intensive investigation in the field of PH curves. This paper tries to summarise research on PH curves over the last years in three major strands: polynomial PH curves, rational PH curves, and Hermite interpolation by PH curves in Euclidean and Minkowski plane and space.

Keywords: Pythagorean hodograph curves, rational offsets, interpolation.

MSC: 53A04; 51N35, 14H45, 68U05

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