Approximating curves and their offsets using biarcs and Pythagorean hodograph quintics

Authors:
Zbynk Šír;Robert Feichtinger;Bert Jüttler
Affiliations:
Institute of Applied Geometry, Johannes Kepler University, Altenberger Street 69, 4040 Linz, Austria;Institute of Applied Geometry, Johannes Kepler University, Altenberger Street 69, 4040 Linz, Austria;Institute of Applied Geometry, Johannes Kepler University, Altenberger Street 69, 4040 Linz, Austria
Venue:
Computer-Aided Design
Year:
2006

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Abstract

This paper compares two techniques for the approximation of the offsets to a given planar curve. The two methods are based on approximate conversion of the planar curve into circular splines and Pythagorean hodograph (PH) splines, respectively. The circular splines are obtained using a novel variant of biarc interpolation, while the PH splines are constructed via Hermite interpolation of C^1 boundary data. We analyze the approximation order of both conversion procedures. As a new result, the C^1 Hermite interpolation with PH quintics is shown to have approximation order 4 with respect to the original curve, and 3 with respect to its offsets. In addition, we study the resulting data volume, both for the original curve and for its offsets. It is shown that PH splines outperform the circular splines for increasing accuracy, due to the higher approximation order.