Geometric Hermite interpolation by spatial Pythagorean-hodograph cubics |
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Authors: | Email author" target="_blank">Francesca?PelosiEmail author Rida?T?Farouki Carla?Manni Alessandra?Sestini |
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Institution: | (1) Dipartimento di Matematica Pura ed Applicata, Università di Padova, via Belzoni 7, 35131 Padova, Italy;(2) Department of Mechanical and Aeronautical Engineering, University of California, Davis, CA 95616, USA;(3) Dipartimento di Matematica, Università di Roma Tor Vergata, via della Ricerca Scientifica, 00133 Roma, Italy;(4) Dipartimento di Energetica, Università di Firenze, via Lombroso 6/17, 50134 Firenze, Italy |
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Abstract: | It is shown that, depending upon the orientation of the end tangents t0,t1 relative to the end point displacement vector p=p1–p0, the problem of G1 Hermite interpolation by PH cubic segments may admit zero, one, or two distinct solutions. For cases where two interpolants exist, the bending energy may be used to select among them. In cases where no solution exists, we determine the minimal adjustment of one end tangent that permits a spatial PH cubic Hermite interpolant. The problem of assigning tangents to a sequence of points p0,...,pn in R3, compatible with a G1 piecewise-PH-cubic spline interpolating those points, is also briefly addressed. The performance of these methods, in terms of overall smoothness and shape-preservation properties of the resulting curves, is illustrated by a selection of computed examples. |
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Keywords: | Pythagorean-hodograph cubics quaternions Hermite interpolation tangent adjustment curvature torsion helix energy integral shape preservation |
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