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A Sobolev gradient method for construction of elastic curves in regular surfaces. (English) Zbl 1224.65041
Summary: Given a sequence of control points and tangent vectors in a regular surface, we treat the problem of constructing a curve that lies in the surface and has minimum total curvature subject to the constraint that it interpolates the control points and tangent vectors. Geodesics and parametric planar curves (nonlinear splines) are included as special cases. The surface is defined implicitly by a smooth function, and the curve is approximated by a discrete set of vertices along with first and second derivative vectors. The nonlinear optimization problem of minimizing curvature subject to the constraints is solved by a variable metric gradient descent method based on Neuberger’s Sobolev gradient theory. We compare different methods for treating the nonlinear constraints.

65D17 Computer-aided design (modeling of curves and surfaces)
65D07 Numerical computation using splines
Full Text: DOI
[1] Brent, R., Algorithms for minimization without derivatives, (1973), Prentice-Hall Englewood Cliffs, NJ · Zbl 0245.65032
[2] J.W. Neuberger, Sobolev gradients and differential equations, Lecture Notes in Mathematics, vol. 1670, Springer, Berlin, 1997. · Zbl 0935.35002
[3] Polak, E., Computational methods in optimization, (1971), Academic Press New York · Zbl 0257.90055
[4] Renka, R.J., Algorithm 828. DNSPLIN1: discrete nonlinear spline interpolation, ACM trans. math. software, 29, 1-11, (2003) · Zbl 1070.65507
[5] Renka, R.J., Constructing fair curves and surfaces with a Sobolev gradient method, Cagd, 21, 137-149, (2004) · Zbl 1069.65565
[6] Renka, R.J., Algorithm 834: glsurf—an interactive surface plotting program using opengl, ACM trans. math. software, 30, 212-217, (2004) · Zbl 1070.65509
[7] Renka, R.J.; Neuberger, J.W., Minimal surfaces and Sobolev gradients, SIAM J. sci. comput., 16, 1412-1427, (1995) · Zbl 0857.35004
[8] Woodford, C.H., Smooth curve interpolation, Bit, 9, 69-77, (1969) · Zbl 0319.65004
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