Gan, Zhiwang; Zhou, Meng Symbolic computation of the orthogonal projection of rational curves onto rational parameterized surfaces. (English) Zbl 1394.65016 Math. Probl. Eng. 2015, Article ID 394246, 11 p. (2015). Summary: This paper focuses on the orthogonal projection of rational curves onto rational parameterized surface. Three symbolic algorithms are developed and studied. One of them, based on regular systems, is able to compute the exact parametric loci of projection. The one based on Gröbner basis can compute the minimal variety that contains the parametric loci. The remaining one computes a variety that contains the parametric loci via resultant. Examples show that our algorithms are efficient and valuable. MSC: 65D18 Numerical aspects of computer graphics, image analysis, and computational geometry 68W30 Symbolic computation and algebraic computation Software:CASA; Epsilon PDFBibTeX XMLCite \textit{Z. Gan} and \textit{M. Zhou}, Math. Probl. Eng. 2015, Article ID 394246, 11 p. (2015; Zbl 1394.65016) Full Text: DOI References: [1] Gilbert, E. G.; Johnson, D. W.; Keerthi, S. S., A fast procedure for computing the distance between complex objects in three-dimensional space, IEEE journal of robotics and automation, 4, 2, 193-203, (1988) [2] Chen, X.-D.; Yong, J.-H.; Wang, G.; Paul, J.-C.; Xu, G., Computing the minimum distance between a point and a NURBS curve, CAD Computer Aided Design, 40, 10-11, 1051-1054, (2008) [3] Limaiem, A.; Trochu, F., Geometric algorithms for the intersection of curves and surfaces, Computers and Graphics, 19, 3, 391-403, (1995) [4] Pegna, J.; Wolter, F.-E., Surface curve design by orthogonal projection of space curves onto free-form surfaces, Transactions of the ASME—Journal of Mechanical Design, 118, 1, 45-52, (1996) [5] Dietz, R.; Hoschek, J.; Jüttler, B., An algebraic approach to curves and surfaces on the sphere and on other quadrics, Computer Aided Geometric Design, 10, 3-4, 211-229, (1993) · Zbl 0781.65009 [6] Ma, Y. L.; Hewitt, W. T., Point inversion and projection for NURBS curve and surface: control polygon approach, Computer Aided Geometric Design, 20, 2, 79-99, (2003) · Zbl 1069.65558 [7] Pottmann, H.; Leopoldseder, S.; Hofer, M., Registration without icp, Computer Vision and Image Understanding, 95, 1, 54-71, (2004) [8] Song, H.-C.; Yong, J.-H.; Yang, Y.-J.; Liu, X.-M., Algorithm for orthogonal projection of parametric curves onto b-spline surfaces, Computer Aided Design, 43, 4, 381-393, (2011) [9] Chernov, N.; Wijewickrema, S., Algorithms for projecting points onto conics, Journal of Computational and Applied Mathematics, 251, 8-21, (2013) · Zbl 1288.65016 [10] Hu, S.-M.; Wallner, J., A second order algorithm for orthogonal projection onto curves and surfaces, Computer Aided Geometric Design, 22, 3, 251-260, (2005) · Zbl 1205.65086 [11] Liu, X.-M.; Yang, L.; Yong, J.-H.; Gu, H.-J.; Sun, J.-G., A torus patch approximation approach for point projection on surfaces, Computer Aided Geometric Design, 26, 5, 593-598, (2009) · Zbl 1205.65092 [12] Oh, Y.-T.; Kim, Y.-J.; Lee, J.; Kim, M.-S.; Elber, G., Efficient point-projection to freeform curves and surfaces, Computer Aided Geometric Design, 29, 5, 242-254, (2012) · Zbl 1250.65038 [13] Busé, L.; Elkadi, M.; Galligo, A., Intersection and self-intersection of surfaces by means of bezoutian matrices, Computer Aided Geometric Design, 25, 2, 53-68, (2008) · Zbl 1172.14346 [14] Wang, D., Elimination Practice: Software Tools and Applications, (2004), London, UK: Imperial College Press, London, UK · Zbl 1099.13047 [15] Huang, Y.; Wang, D., Computing intersection and self-intersection loci of parametrized surfaces using regular systems and Gröbner bases, Computer Aided Geometric Design, 28, 9, 566-581, (2011) · Zbl 1247.65022 [16] Piegl, L.; Tiller, W., The NURBS Book, (2012), Berlin, Germany: Springer, Berlin, Germany [17] Wang, D., Computing triangular systems and regular systems, Journal of Symbolic Computation, 30, 2, 221-236, (2000) · Zbl 1007.65039 [18] Buchberger, B., Gröbner bases: a short introduction for systems theorists, Computer Aided Systems Theory—EUROCAST 2001. Computer Aided Systems Theory—EUROCAST 2001, Lecture Notes in Computer Science, 2178, 1-19, (2001), Berlin, Germany: Springer, Berlin, Germany · Zbl 1023.68882 [19] Mishra, B., Algorithmic Algebra, (1993), New York, NY, USA: Springer, New York, NY, USA · Zbl 0804.13009 [20] Cox, D.; Little, J.; O’Shea, D., Ideals, Varieties, and Algorithms: An Introduction to Computational Algebraic Geometry and Commutative Algebra, (2007), Springer · Zbl 1118.13001 [21] van Hoeij, M., Rational parametrizations of algebraic curves using a canonical divisor, Journal of Symbolic Computation, 23, 2-3, 209-227, (1997) · Zbl 0878.68073 [22] Wang, D., Elimination Methods. Elimination Methods, Texts and Monographs in Symbolic Computation, (2001), Berlin, Germany: Springer, Berlin, Germany [23] Gan, Z.; Zhou, M., Computing the orthogonal projection of rational curves onto rational parameterized surface by symbolic methods, Mathematical Software—ICMS 2014. Mathematical Software—ICMS 2014, Lecture Notes in Computer Science, 8592, 261-268, (2014), Berlin, Germany: Springer, Berlin, Germany · Zbl 1437.14062 [24] Winkler, F., Polynomial Algorithms in Computer Algebra, (1996), Vienna, Austria: Springer, Vienna, Austria · Zbl 0853.12003 [25] Sendra, J. R.; Winkler, F.; Pérez-Díaz, S., Rational Algebraic Curves: A Computer Algebra Approach, (2008), Berlin, Germany: Springer, Berlin, Germany · Zbl 1129.14083 [26] Warkentin, A.; Ismail, F.; Bedi, S., Comparison between multi-point and other 5-axis tool positioning strategies, International Journal of Machine Tools and Manufacture, 40, 2, 185-208, (2000) 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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.