Abstract
In the 19th century, the French geometer Charles Pierre Dupin discovered a non-spherical surface with circular lines of curvature. He called it a cyclide in his book,Applications de Geometrie published in 1822. Recently, cyclides have been revived for use as surface patches in computer aided geometric design (CAGD). Other applications of cyclides in CAGD are possible (e.g., variable radius blending) and require a deep understanding of the geometry of the cyclide. We resurrect the geometric descriptions of the cyclide found in the classical papers of James Clerk Maxwell and Arthur Cayley. We present a unified perspective of their results and use them to devise effective algorithms for synthesizing cyclides. We also discuss the morphology of cyclides and present a new classification scheme.
Similar content being viewed by others
References
Casey J (1871) On cyclides and spheroquartics. Philos Trans R Soc Dublin, pp 585–721
Cayley A (1873) On the cyclide. Q J Pure Appl Math 12:148–163
Chandru V, Dutta D, Hoffmann CM (1989) Variable radius blending using Dupin cyclides. CSD-TR-851, Comput Sci Dep, Purdue Univ (January 1989)
Darboux G (1887) La theorie generale des surfaces. Gautheir-Villars, Paris
Dupin C (1822) Applications de Geometrie et de Mechanique. Bachelier, Paris
Hilbert D, Cohn-Vossen S (1932) Geometry and the imagination. Chelsea Publ, New York (reprinted 1983), p 218
Hoffmann CM (1988) Variable radius blending using cyclides, NSF-IFIP Workshop on Geometric Modeling, Rensselaerville, NY (September 1988)
Knapman K (1986) Dupins cyclide and the cyclide patch. Document No, UKSC 156, IBM UK Scientific Centre, Winchester, England
Maxwell JC (1868) On the cyclide. Q J Pure Appl Math 9:111–126
Martin RR (1983) Principal patches — a new class of surface patch. Proc Eurographics 1983, 47, North Holland, Amsterdam
Martin RR, de Pont JJ, Sharrock TJ (1986) Cyclide surfaces for computer-aided design. In: Gregory J (ed) (1984) Mathematics of Surfaces. Conf Proc, Institute of Mathematics and its Applications, Oxford Univ Press, Oxford
McLean D (1984) A method for generating surfaces as a composite of cyclide patches. Comput J 28(4):433–438
Nutbourne AW, Martin RR (1988) Differential geometry applied to curve and surface design, vol 1. Foundations. Ellis Horwood Publ, p 223
Pratt MJ (1988) Applications of cyclide surfaces in geometric modeling. Proc 3rd IMA Conf on the Mathematics of Surfaces, Oxford (September 1988) Oxford Univ Press (in preparation)
Salmon G (1915) A treatise on the analytic geometry of three dimensions. Longmans Green, London
Struik DJ (1961) Lectures on classical differential geometry, 2nd ed. Addison-Wesley, Dover (reprint 1988)
Woods FS (1922) Higher geometry, ginn and company
Author information
Authors and Affiliations
Additional information
Chandru was supported in part by grants from the ONR under University Research Initiative grant number N00014-86-K-0689 and from NSF grant number DMC 88-07550
Dutta received support from the National Science Foundation grant number DMC 88-07550
Hoffmann was supported in part by NSF grants CCR-8619817 and DMC 88-07550 and ONR contract N0014-86-K-0465, and a grant from the AT & T Foundation
Rights and permissions
About this article
Cite this article
Chandru, V., Dutta, D. & Hoffmann, C.M. On the geometry of Dupin cyclides. The Visual Computer 5, 277–290 (1989). https://doi.org/10.1007/BF01914786
Issue Date:
DOI: https://doi.org/10.1007/BF01914786