Abstract
We study Bézier-like formulas with weights in geometric algebra for parametrizing a special class of rational surfaces in isotropic 3-space. These formulas are useful for constructing isotropic-Möbius invariant surfaces that are dual to rational offset surfaces in euclidean 3-space. Our focus is on bilinear Clifford-Bézier patches. We derive their implicitization formula and characterize them as patches on special quartic surfaces called isotropic cyclides. Finally we present one modeling application with rational surfaces admitting rational offsets.
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Ablamowicz, R., Fauser, B.: A Maple 10 Package for Clifford Algebra Computations, Version 10 (2007), http://math.tntech.edu/rafal/cliff10
Ciliberto, C., van der Geer, G.: Andreotti–Mayer loci and the Schottky problem. Documenta Mathematica 13, 453–504 (2008)
Dorst, L., Fontijne, D., Mann, S.: Geometric Algebra to Computer Science. Morgan-Kaufmann (2007)
Goldman, R.: A Homogeneous Model for Three-Dimensional Computer Graphics Based on the Clifford Algebra for \(\mbox{I\!R}^3\). In: Dorst, L., Lasenby, J. (eds.) Guide to Geometric Algebra in Practice, pp. 329–352 (2011)
Krasauskas, R., Peternell, M.: Rational offset surfaces and their modeling applications. In: Emiris, I.Z., Sottile, F., Theobald, T. (eds.) IMA Volume 151: Nonlinear Computational Geometry, pp. 109–135 (2010)
Krasauskas, R.: Branching blend of natural quadrics based on surfaces with rational offsets. Computer Aided Geometric Design 25, 332–341 (2008)
Krasauskas, R., Zube, S.: Bezier-like parametrizations of spheres and cyclides using geometric algebra. In: Guerlebeck, K. (ed.) Proceedings of 9th International Conference on Clifford Algebras and their Applications in Mathematical Physics, Weimar, Germany (2011)
Perwass, C.: Geometric Algebra with Applications in Engineering. Series: Geometry and Computing, vol. 4. Springer (2009)
Peternell, M., Pottmann, H.: A Laguerre geometric approach to rational offsets. Computer Aided Geometric Design 15, 223–249 (1998)
Pottmann, H., Grohs, P., Mitra, N.J.: Laguerre Minimal Surfaces, Isotropic Geometry and Linear Elasticity. Advances in Computational Mathematics 31, 391–419 (2009)
Pottmann, H., Peternell, M.: Applications of Laguerre geometry in CAGD. Computer Aided Geometric Design 15, 165–186 (1998)
Pottmann, H., Shi, L., Skopenkov, M.: Darboux cyclides and webs from circles. Computer Aided Geometric Design 29, 77–97 (2012)
Zube, S.: A circle represenatation using complex and quaternion numbers. Lithuanian Journal of Mathematics 46, 298–310 (2006)
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Krasauskas, R., Zubė, S., Cacciola, S. (2014). Bilinear Clifford-Bézier Patches on Isotropic Cyclides. In: Floater, M., Lyche, T., Mazure, ML., Mørken, K., Schumaker, L.L. (eds) Mathematical Methods for Curves and Surfaces. MMCS 2012. Lecture Notes in Computer Science, vol 8177. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-54382-1_17
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DOI: https://doi.org/10.1007/978-3-642-54382-1_17
Publisher Name: Springer, Berlin, Heidelberg
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