Abstract
Dupin cyclides are algebraic surfaces of order three and four whose lines of curvature are circles. These surfaces have a variety of interesting properties and are aesthetic from a geometric and algebraic viewpoint. Besides their special property with respect to lines of curvature they appear as envelopes of one-parameter families of spheres in a twofold way. In the present article we study two families of canal surfaces with rational lines of curvature and rational principal curvatures, which contain the Dupin cyclides of order three and four as special instances in each family. The surfaces are constructed as anticaustics with respect to parallel illumination and reflection at tangent planes of curves on a cylinder of rotation.
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References
Allen, S., Dutta, D.: Supercyclides and blending. Comput. Aided Geom. Design 14, 637–651 (1997)
Biard, L., Farouki, R.T., Szafran, N.: Construction of rational surface patches bounded by lines of curvature. Comput. Aided Geom. Design 27, 359–371 (2010)
do Carmo, M.: Differential Geometry of Curves and Surfaces. Prentice-Hall, Englewood Cliffs (1976)
Darboux, G.: Sur une classe remarquable de courbes et de surfaces algebrique, 2nd edn., Gauthier-Villars, Paris (1896)
Degen, W.L.F.: Nets with Plane Silhouettes. In: The Mathematics of Surfaces V, Design and Application of Curves and Surfaces, pp. 117–133. Oxford Univ. Press, Oxford (1994)
Dupin, C.: Applications de Geometrie et de Mechanique. Bachelier, Paris (1822)
Krasauskas, R.: Branching blend of natural quadrics based on surfaces with rational offsets. Comput. Aided Geom. Design 25, 332–341 (2008)
Krasauskas, R., Mäurer, C.: Studying cyclides with Laguerre geometry. Comput. Aided Geom. Design 17, 101–126 (1999)
Peternell, M., Pottmann, H.: A Laguerre geometric approach to rational offsets. Comput. Aided Geom. Design 15, 223–249 (1998)
Pottmann, H., Peternell, M.: Applications of Laguerre geometry in CAGD. Comput. Aided Geom. Design 15, 165–186 (1998)
Pottmann, H.: Rational curves and surfaces with rational offsets. Comput. Aided Geom. Design 12, 175–192 (1995)
Pottmann, H., Wagner, M.: Principal Surfaces. In: The Mathematics of Surfaces VII, pp. 337–362. Information Geometers Ltd. (1998)
Pratt, M.J.: Cyclides in computer aided geometric design. Comput. Aided Geom. Design 7, 221–242 (1990)
Pratt, M.J.: Quartic supercyclides I: Basic theory. Comput. Aided Geom. Design 14, 671–692 (1997)
Pratt, M.J.: Cyclides in computer aided geometric design II. Comput. Aided Geom. Design 12, 131–152 (1995)
Srinivas, Y.L., Kumar, V., Dutta, D.: Surface design using cyclide patches. Computer-Aided Design 28, 263–276 (1996)
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Peternell, M. (2012). Generalized Dupin Cyclides with Rational Lines of Curvature. In: Boissonnat, JD., et al. Curves and Surfaces. Curves and Surfaces 2010. Lecture Notes in Computer Science, vol 6920. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-27413-8_35
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DOI: https://doi.org/10.1007/978-3-642-27413-8_35
Publisher Name: Springer, Berlin, Heidelberg
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