Constructing fair curves and surfaces with a Sobolev gradient method. (English) Zbl 1069.65565

Summary: We have devised a new method for constructing discrete approximations to fair curves and surfaces by directly minimizing an arbitrarily selected fairness functional subject to geometric constraints. The nonlinear optimization problem is solved efficiently by a Sobolev gradient method. We first describe the method in general terms and then present results which demonstrate its effectiveness for constructing minimum variation curves which interpolate specified control points, tangent vectors, and/or curvature vectors.


65D18 Numerical aspects of computer graphics, image analysis, and computational geometry
65D17 Computer-aided design (modeling of curves and surfaces)
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[1] Asaturyan, S.; Costantini, P.; Manni, C., Local shape-preserving interpolation by space curves, IMA J. numer. anal., 21, 301-325, (2001) · Zbl 0976.65007
[2] Goodman, T.N.T.; Ong, B.H., Shape-preserving interpolation by space curves, Computer aided geometric design, 15, 1-17, (1997) · Zbl 0891.68115
[3] Goodman, T.N.T.; Ong, B.H.; Sampoli, M.L., Automatic interpolation by fair, shape preserving, G2 space curves, Computer-aided design, 30, 813-822, (1998) · Zbl 1083.65502
[4] Kaklis, P.D.; Karavelas, M.I., Shape-preserving interpolation in \( R\^{}\{3\}\), IMA J. numer. anal., 17, 373-419, (1997) · Zbl 0928.65019
[5] Moreton, H.P.; Séquin, C.H., Minimum variation curves and surfaces for computer-aided geometric design, (), 123-159 · Zbl 0834.41003
[6] Neuberger, J.W., Sobolev gradients and differential equations, Lecture notes in mathematics, vol. 1670, (1997), Springer Berlin · Zbl 0870.47029
[7] Neuberger, J.W.; Renka, R.J., Sobolev gradients and the ginzburg – landau functional, SIAM J. sci. comput., 20, 582-590, (1998) · Zbl 0920.35059
[8] Polak, E., Computational methods in optimization, (1971), Academic Press New York · Zbl 0257.90055
[9] Raydan, M., The Barzilai and Borwein gradient method for the large scale unconstrained minimization problem, SIAM J. optim., 7, 26-33, (1997) · Zbl 0898.90119
[10] Renka, R.J.; Neuberger, J.W., Minimal surfaces and Sobolev gradients, SIAM J. sci. comput., 16, 1412-1427, (1995) · Zbl 0857.35004
[11] Roulier, J.; Rando, T., Measures of fairness for curves and surfaces, (), 75-120 · Zbl 0834.41002
[12] Schneider, R.; Kobbelt, L., Generating fair meshes with G1 boundary conditions, (), 251-261
[13] Welch, W.; Witkin, A., Free-form shape design using triangulated surfaces, (), 247-256
[14] Woodford, C.H., Smooth curve interpolation, Bit, 9, 69-77, (1969) · Zbl 0319.65004
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