# zbMATH — the first resource for mathematics

Numerical stability of surface implicitization. (English) Zbl 1124.65017
Summary: For a numerically given parametrization we cannot compute an exact implicit equation, just an approximate one. We introduce a condition number to measure the worst effect on the solution when the input data is perturbed by a small amount.

##### MSC:
 65D17 Computer-aided design (modeling of curves and surfaces)
##### Keywords:
condition number; curves; numerical examples; implicit equation
Full Text:
##### References:
 [1] Alonso, C.; Gutierrez, J.; Recio, T., An implicitization algorithm with fewer variables, Comput. aided geom. design, 12, 251-258, (1995) · Zbl 0875.68835 [2] Berry, T.G.; Patterson, R.R., Implicitization and parametrization of nonsingular cubic surfaces, Comput. aided geom. design, 18, 8, 723-738, (2001) · Zbl 0983.68221 [3] Buchberger, B., Applications of Gröbner bases in nonlinear computational geometry, (), 52-80 [4] Busé, L., Residual resultant over the projective plane and the implicitization problem, (), 48-55, (electronic) · Zbl 1356.14057 [5] Castro, D.; Montaña, J.L.; Pardo, L.M.; San Martín, J., The distribution of condition numbers of rational data of bounded bit length, Found. comput. math., 2, 1, 1-52, (2002) · Zbl 1011.65017 [6] Chen, F., Approximate implicitization of rational surfaces, (), 57-65 · Zbl 1027.65022 [7] Corless, R.M.; Giesbrecht, M.W.; Kotsireas, I.S.; Watt, S.M., Numerical implicitization of parametric hypersurfaces with linear algebra, (), 174-183 · Zbl 1042.65020 [8] Dokken, T., 1997. Aspects of intersection algorithms and approximation. Ph.D. Thesis, University of Oslo, Norway · Zbl 0883.65127 [9] Dokken, T., Approximate implicitization, (), 81-102 · Zbl 0989.65019 [10] Dokken, T.; Kellermann, H.K.; Tegnander, C., An approach to weak approximate implicitization, (), 103-112 · Zbl 0989.65020 [11] Dokken, T.; Thomassen, J.B., Overview of approximate implicitization, (), 169-184 · Zbl 1039.65012 [12] Elkadi, M.; Mourrain, B., Residue and implicitization problem for rational surfaces, Appl. algebra engrg. comm. comput., 14, 5, 361-379, (2004) · Zbl 1058.14073 [13] Marco, A.; Martínez, J.J., Implicitization of rational surfaces by means of polynomial interpolation, Comput. aided geom. design, 19, 5, 327-344, (2002) · Zbl 0995.68140 [14] Sederberg, T.W., Techniques for cubic algebraic surfaces 2, IEEE computer graphics and applications, 10, 12-21, (1990) [15] Winkler, J.R., Numerical and algebraic properties of Bernstein basis resultant matrices, (), 107-118 · Zbl 1065.65062 [16] Winkler, J.R.; Goldman, R.N., The Sylvester resultant matrix for Bernstein polynomials, (), 407-416 · Zbl 1043.41004 [17] Zheng, J.; Sederberg, T.W.; Chionh, E.-W.; Cox, D.A., Implicitizing rational surfaces with base points using the method of moving surfaces, (), 151-168 · Zbl 1058.14074
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.