×

Branching blend of natural quadrics based on surfaces with rational offsets. (English) Zbl 1172.65335

Summary: A new branching blend between two natural quadrics (circular cylinders/cones or spheres) in many positions is proposed. The blend is a ring shaped patch of a PN surface (surface with rational offset) parametrized by rational bivariant functions of degree (6,3). The general theory of PN surfaces is developed using Laguerre geometry and a universal rational parametrization of the Blaschke cylinder. The construction is extended via inversion to a PN branching blend of degree (8,4) between Dupin cyclide and a natural quadric.

MSC:

65D17 Computer-aided design (modeling of curves and surfaces)
53A05 Surfaces in Euclidean and related spaces
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Boehm, W., On cyclides in geometric modeling, Computer Aided Geometric Design, 7, 243-255 (1990) · Zbl 0712.65009
[2] Cox, D., Krasauskas, R., Mustaţǎ, M., 2003. Universal rational parametrizations and toric varieties. Topics in Algebraic Geometry and Geometric Modeling, Contemporary Mathematics, vol. 334, pp. 241-265; Cox, D., Krasauskas, R., Mustaţǎ, M., 2003. Universal rational parametrizations and toric varieties. Topics in Algebraic Geometry and Geometric Modeling, Contemporary Mathematics, vol. 334, pp. 241-265 · Zbl 1080.14539
[3] Degen, W., Cyclides, (Farin, G.; Hoschek, J.; Kim, M.-S., Handbook of Computer Aided Geometric Design (2002), Elsevier Science), 575-602 · Zbl 1003.68179
[4] Farouki, R. T.; Pottmann, H., Polynomial and rational Pythagorean-hodograph curves reconciled, (Mullineux, G., The Mathematics of Surfaces VI (1996), Oxford Univ. Press), 355-378 · Zbl 0878.68120
[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] Kazakevičiūtė, M., 2005a. Blending of natural quadrics with rational canal surfaces. PhD Thesis, Vilnius University; Kazakevičiūtė, M., 2005a. Blending of natural quadrics with rational canal surfaces. PhD Thesis, Vilnius University · Zbl 1083.53019
[7] Kazakevičiūtė, M., Classification of pairs of natural quadrics from the point of view of Laguerre geometry, Lithuanian Mathematical Journal, 45, 63-84 (2005) · Zbl 1083.53019
[8] Krasauskas, R., Bézier patches on almost toric surfaces, (Elkadi, M.; Mourrain, B.; Piene, R., Algebraic Geometry and Geometric Modeling. Algebraic Geometry and Geometric Modeling, Mathematics and Visualization Series (2006), Springer), 135-150 · Zbl 1110.14307
[9] Krasauskas, R., Minimal rational parametrizations of canal surfaces, Computing, 79, 281-290 (2007) · Zbl 1112.14317
[10] Krasauskas, R.; Kazakevičiūtė, M., Universal rational parametrizations and spline curves on toric surfaces, (Computational Methods for Algebraic Spline Surfaces, ESF Exploratory Workshop (2005), Springer), 213-231 · Zbl 1065.65019
[11] Krasauskas, R., Zubė, S., 2007. Canal surfaces defined by quadratic families of spheres. In: Proceedings of the COMPASS II Workshop, pp. 138-150; Krasauskas, R., Zubė, S., 2007. Canal surfaces defined by quadratic families of spheres. In: Proceedings of the COMPASS II Workshop, pp. 138-150
[12] Peternell, M., 1997. Rational parametrizations for envelopes of quadric families. Ph.D. Thesis, Institute of Geometry, University of Technology, Vienna; Peternell, M., 1997. Rational parametrizations for envelopes of quadric families. Ph.D. Thesis, Institute of Geometry, University of Technology, Vienna
[13] Peternell, M.; Pottmann, H., Computing rational parametrizations of canal surfaces, Journal of Symbolic Computation, 23, 255-266 (1997) · Zbl 0877.68116
[14] Pottmann, H., Rational curves and surfaces with rational offsets, Computer Aided Geometric Design, 12, 175-192 (1993) · Zbl 0872.65011
[15] Pottmann, H.; Peternell, M., Application of Laguerre geometry in CAGD, Computer Aided Geometric Design, 15, 165-186 (1998) · Zbl 0996.65018
[16] Pratt, M. J., Cyclides in computer aided geometric design, Computer Aided Geometric Design, 7, 221-242 (1990) · Zbl 0712.65008
[17] Rossignac, J.R., 1987. Constraints in constructive solid geometry. In: Proceedings of the 1986 Workshop on Interactive 3D Graphics. Chapel Hill, North Carolina, 1987, pp. 93-110; Rossignac, J.R., 1987. Constraints in constructive solid geometry. In: Proceedings of the 1986 Workshop on Interactive 3D Graphics. Chapel Hill, North Carolina, 1987, pp. 93-110
[18] Shene, C.-K., Blending two cones with Dupin cyclides, Computer Aided Geometric Design, 15, 643-673 (1998) · Zbl 0905.68151
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.