Abstract
We present two numerical methods to approximate the shortest path or a geodesic between two points on a three-dimensional parametric surface. The first one consists of minimizing the path length, working in the parameter domain, where the approximation class is composed of Bézier curves. In the second approach, we consider Bézier surfaces and their control net. The numerical implementation is based on finding the shortest path on the successive control net subdivisions. The convergence property of the Bézier net to the surface gives an approximation of the required shortest path. These approximations, also called pseudo-geodesics, are then applied to the creation of pseudo-geodesic meshes. Experimental results are also provided.
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References
C. Duflos, P. Meheut and J.C. Weill, Calcul de géodésiques par une méthode de projection, Research Report No. 0907, Inria-Roquencourt (1988).
R. Kimmel, A. Amir and A.M. Bruckstein, Finding shortest paths on surfaces, in: Curves and Surfaces in Geometric Design, eds. P.-J. Laurent, A. LeMéhauté and L.L. Schumaker (A.K. Peters, Wellesley, MA, 1994) pp. 259–268.
T. Maekawa, Computation of shortest paths on free-form parametric surfaces, J. Mechanical Design 118 (1996) 499–508.
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Pham-Trong, V., Szafran, N. & Biard, L. Pseudo-geodesics on three-dimensional surfaces and pseudo-geodesic meshes. Numerical Algorithms 26, 305–315 (2001). https://doi.org/10.1023/A:1016617010088
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DOI: https://doi.org/10.1023/A:1016617010088