Dual representation of spatial rational Pythagorean-hodograph curves |
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Authors: | Jernej Kozak Marjeta Krajnc Vito Vitrih |
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Affiliation: | 1. FMF, University of Ljubljana, Jadranska 19, Ljubljana, Slovenia;2. IMFM, Jadranska 19, Ljubljana, Slovenia;3. IAM, University of Primorska, Muzejski trg 2, Koper, Slovenia;4. FAMNIT, University of Primorska, Glagoljaška 8, Koper, Slovenia |
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Abstract: | In this paper, the dual representation of spatial parametric curves and its properties are studied. In particular, rational curves have a polynomial dual representation, which turns out to be both theoretically and computationally appropriate to tackle the main goal of the paper: spatial rational Pythagorean-hodograph curves (PH curves). The dual representation of a rational PH curve is generated here by a quaternion polynomial which defines the Euler–Rodrigues frame of a curve. Conditions which imply low degree dual form representation are considered in detail. In particular, a linear quaternion polynomial leads to cubic or reparameterized cubic polynomial PH curves. A quadratic quaternion polynomial generates a wider class of rational PH curves, and perhaps the most useful is the ten-parameter family of cubic rational PH curves, determined here in the closed form. |
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Keywords: | Pythagorean-hodograph Rational curves Dual parametric curve form Frenet frame Quaternions |
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