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Rational surfaces with linear normals and their convolutions with rational surfaces. (English) Zbl 1095.53006

The authors discuss geometric properties of polynomial (or rational) parametrized surfaces with linear fields of normal vectors. It is shown that these are projectively dual to graphs of bivariate polynomials (or rational functions). This dual representation is applied to prove that the convolution of rational surfaces with linear normal vector fields with general rational surfaces is again rational.

MSC:

53A05 Surfaces in Euclidean and related spaces
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[1] Bajaj, C.; Kim, M.-S., Generation of configuration space obstacles: The case of a moving algebraic curve, Algorithmica, 4, 2, 157-172 (1989) · Zbl 0672.68009
[2] Farin, G.; Hoschek, J.; Kim, M.-S., Handbook of Computer Aided Geometric Design (2002), Elsevier: Elsevier Amsterdam · Zbl 1003.68179
[3] Farouki, R. T., Minkowski combination of complex sets—geometry, algorithms and applications, (Lyche, T.; Mazure, M.-L.; Schumaker, L. L., Curve and Surface Design: Saint Malo, 2002 (2003), Nashboro Press), 123-146 · Zbl 1047.65029
[4] Jüttler, B., Triangular Bézier surface patches with a linear normal vector field, (The Mathematics of Surfaces VIII (1998), Information Geometers), 431-446 · Zbl 0959.65038
[5] Jüttler, B.; Sampoli, M. L., Hermite interpolation by piecewise polynomial surfaces with rational offsets, Computer Aided Geometric Design, 17, 361-385 (2000) · Zbl 0938.68123
[6] Kaul, A.; Farouki, R. T., Computing Minkowski sums of planar curves, Int. J. Computat. Geom. Appl., 5, 413-432 (1995) · Zbl 0854.68102
[7] Kohler, K.; Spreng, M., Fast computation of the C-space of convex 2D algebraic objects, Int J. Robotics Res., 14, 6, 590-608 (1995)
[8] Landsmann, G.; Schicho, J.; Winkler, F., The parametrization of canal surfaces and the decomposition of polynomials into a sum of two squares, J. Symbolic Comput., 32, 1-2, 119-132 (2001) · Zbl 1079.12501
[9] Lee, I.-K.; Kim, M.-S.; Elber, G., Polynomial/rational approximation of Minkowski sum boundary curves, Graphical Models, 60, 2, 136-165 (1998)
[10] Lee, I.K., Kim, M.-S., Elber, G., 1998b. The Minkowski sum of 2D curved objects. In: Proceedings of Israel-Korea Bi-National Conference on New Themes in Computerized Geometrical Modeling, Tel-Aviv University, pp. 155-164; Lee, I.K., Kim, M.-S., Elber, G., 1998b. The Minkowski sum of 2D curved objects. In: Proceedings of Israel-Korea Bi-National Conference on New Themes in Computerized Geometrical Modeling, Tel-Aviv University, pp. 155-164
[11] Li, A.-M.; Simon, U.; Zhao, G., Global Affine Differential Geometry of Hypersurfaces (1993), de Gruyter: de Gruyter Berlin · Zbl 0808.53002
[12] Lü, W., Rationality of the offsets to algebraic curves and surfaces, Appl. Math.-JCU, 9B, 265-278 (1994) · Zbl 0814.14048
[13] Mühlthaler, H.; Pottmann, H., Computing the Minkowski sum of ruled surfaces, Graphical Models, 65, 369-384 (2003) · Zbl 1069.68110
[14] Peternell, M.; Pottmann, H., A Laguerre geometric approach to rational offsets, Computer Aided Geometric Design, 15, 223-249 (1998) · Zbl 0903.68190
[15] Peternell, M.; Manhart, F., The convolution of a paraboloid and a parametrized surface, J. Geometry Graph., 7, 157-171 (2003) · Zbl 1056.51016
[16] Pottmann, H., General offset surfaces, Neural Parallel Scientific Comput., 5, 55-80 (1997)
[17] Ramkumar, G.D., 1996. An algorithm to compute the Minkowski sum outer-face of two simple polygons. In: Proc. ACM Symposium on Computational Geometry; Ramkumar, G.D., 1996. An algorithm to compute the Minkowski sum outer-face of two simple polygons. In: Proc. ACM Symposium on Computational Geometry
[18] Sampoli, M.L., 2005. Computing the convolution and the Minkowski sum of surfaces. In: Proceedings of the Spring Conference on Computer Graphics, Comenius University, Bratislava. ACM Siggraph, in press; Sampoli, M.L., 2005. Computing the convolution and the Minkowski sum of surfaces. In: Proceedings of the Spring Conference on Computer Graphics, Comenius University, Bratislava. ACM Siggraph, in press
[19] Schicho, J., Rational parametrization of surfaces, J. Symbolic Comput., 26, 1-30 (1998) · Zbl 0924.14027
[20] Schicho, J., Proper parametrization of real tubular surfaces, J. Symbolic Comput., 30, 583-593 (2000) · Zbl 1031.14029
[21] Sendra, J. R.; Sendra, J., Rationality analysis and direct parametrization of generalized offsets to quadrics, Appl. Algebra Engrg. Commun. Comput., 11, 2, 111-139 (2000) · Zbl 1053.14068
[22] Seong, J.-K., Kim, M.-S., Sugihara, K., 2002. The Minkowski sum of two simple surfaces generated by slope-monotone closed curves. In: Proceedings of Geometric Modeling and Processing, 2002, Japan, pp. 33-42; Seong, J.-K., Kim, M.-S., Sugihara, K., 2002. The Minkowski sum of two simple surfaces generated by slope-monotone closed curves. In: Proceedings of Geometric Modeling and Processing, 2002, Japan, pp. 33-42
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