Construction of Bézier rectangles and triangles on the symmetric dupin horn cyclide by means of inversion

Computer Aided Geometric Design

Volumn 14, Issue 4, 1997, Pages 349-375

  Albrecht, Gudrun ^a,b   Degen, Wendelin L F ^c  

^a Rheinland Pfälzische Technische Universität Kaiserslautern Landau   (Germany)

^b TECHNICAL UNIVERSITY OF MUNICH   (Germany)

^c UNIVERSITY OF STUTTGART   (Germany)

Author keywords

Dupin cyclide; Inversion; Rational rectangular and triangular B zier representation

Indexed keywords

ALGEBRA; ALGORITHMS; COMPUTER AIDED DESIGN; INVERSE PROBLEMS;

BEZIER REPRESENTATION; DUPIN HORN CYCLIDE;

COMPUTATIONAL GEOMETRY;

EID: 0031140640     PISSN: 01678396     EISSN: None     Source Type: Journal    
DOI: 10.1016/S0167-8396(97)00002-2     Document Type: Article

References (41)

1. Brakhage, K.-H. (1993), A short note on cyclide intersections, distributed at the Third SIAM Conference on Geometric Design, Tempe.
2. Bromwich, T.J.T.A. (1906), Quadratic Forms and their Classification by Means of Invariant-Factors, Hafner Publ. Co., New York.
3. Cayley, A. (1873), On the cyclide, Quart. J. Pure Appl. Math. 12, 148-163.
4. Casey, J. (1871), On cyclides and sphero-quartics, Phil. Trans. Roy. Soc. 161, 585-721.
5. Chandru, V., Dutta, D. and Hoffmann, Ch.M. (1990), Variable radius blending using Dupin cyclides, in: Wozny, M.J., Turner, J.U. and Preiss, K., eds., Geometric Modeling for Product Engineering, North-Holland, Amsterdam, 39-57.
6. Darboux, G. (1872), Mémoire sur les surfaces cyclides, Ann. Ec. Norm. (2) 1, 273-292.
7. Degen, W.L.F. (1986), Die zweifachen Blutelschen Kegelschnittflächen, Manuscripta Math, 55, 9-38.
8. Degen, W.L.F. (1994a), Generalized cyclides for use in CAGD, in: Bowyer, A., ed., Mathematics of Surfaces IV, Clarendon Press, Oxford.
9. Degen, W.L.F. (1994b), Nets with plane silhouettes, in: Fisher, R.B., ed., Design and Application of Curves and Surfaces, Mathematics of Surfaces V, Clarendon Press, Oxford.
10. Degen, W.L.F. (1994c), The type of triangular Bézier surfaces, in: The Mathematics of Surfaces VI (Proc. of the VI, IMA Conference on the Mathematics of Surfaces, Brunei University, September 1994) to appear.
11. Doehlemann, K. (1908), Geometrische Transformationen, Zweiter Teil, Die quadratischen und höheren, birationalen Punkttransformationen, Göschen'sche Verlagshandlung, Leipzig.
12. Dupin, Ch. (1822), Applications de Géometrie et de Me'chanique à la Marine et aux Points et Chaussées, Bachelier, Paris.
13. Srinivas, V.L. and Dutta, D. (1992), Motion Planning in Three Dimensions Using Cyclides, in: Kunii, T.L., ed., Visual Computing, Springer, Tokyo, 781-790.
14. Srinivas, Y.L. and Dutta, D. (1994), Intuitive procedure for constructing geometrically complex objects using cyclides, Computer-Aided Design 26, 327-335.
15. Fladt, K. and Baur, A. (1975), Analytische Geometrie spezieller Flächen und Raumkurven, Vieweg, Braunschweig.
16. Franke, R., Hagen, H. and Nielson, G.M. (1994), Least squares surface approximation to scattered data using multiquadric functions, Adv. in Comput. Math. 2, 81-99.
17. Franke, R., Hagen, H. and Nielson, G.M. (1993), Repeated knots in least squares multiquadric functions, Technical Report NPS-MA-94-004, Naval Postgraduate School, Monterey CA.
18. Hajós, G. (1970), Einführung in die Geometrie, Teubner, Leipzig.
19. Hirst, A.E. (1990), Blending cones and planes by using cyclides of Dupin, Bull. Inst. Math. Appl. 26, 41-46.
20. Johnstone, J.K. and Shene, C.-K. (1994), Dupin cyclides as blending surfaces for cones, in: Fisher, R.B., ed., Design and Application of Curves and Surfaces, Mathematics of Surfaces V. Clarendon Press, Oxford.
21. Johnstone, J.K. (1993), A new intersection algorithm for cyclides and swept surfaces using circle decomposition, Computer Aided Geometric Design 10, 1-24.
22. Kaps, M. (1991), Teilflächen einer Dupinschen Zyklide in Bézierdarstellung, Diss. Universität Braunschweig.
23. Kommerell, K. (1918), Elementargeometrische Behandlung der Dupinschen Zyklide, Zeitschrift für math, und nat. Unterricht 49, 79-95.
24. Kubota, K.K. (1972), Pythagorean triples in unique factorization domains, Amer. Math. Monthly 79, 503-505.
25. Liouville, J. (1847), J. Math. Pures Appl. (1) 12, 265-290.
26. Lockwood, E.H. (1967), A Book of Curves, Univ. Press, Cambridge.
27. Martin, R.R. (1982), Principal patches for computational geometry, Diss. University of Cambridge.
28. Martin, R.R., Pont, J. de and Sharrock, T.J. (1986), Cyclide surfaces in computer aided design, in: Gregory, J. A., ed., The Mathematics of Surfaces, Clarendon Press, Oxford, 253-267.
29. Maxwell, J.C. (1868), On the cyclide, Quart. J. Pure Appl. Math. 9, 111-126.
30. Niebuhr, J. (1992), Eigenschaften der Darstellung insbesondere degenerierter Quadriken mittels Dreiecks-Bézier-Flächen, Diss. Universität, Braunschweig.
31. Pascal, E. and Timerding, H.E. (1922), Repertorium der Höheren Mathematik, Zweiter Band, Zweite Hafte, Teubner, Leipzig und Berlin.
32. Pratt, M.J. (1990), Cyclides in computer aided geometric design. Computer Aided Geometric Design 7, 221-242.
33. Pratt, M.J. (1995), Cyclides in computer aided geometric design II, Computer Aided Geometric Design 12, 131-152.
34. Pratt, M.J. (1994), Quartic supercyclides I: Basic theory, in: The Mathematics of Surfaces VI (Proc. of the VI, IMA Conference on the Mathematics of Surfaces, Brunei University, September 1994) to appear.
35. Roberts, W. (1861), Construction géométrique des surfaces ayant pour lieux des centres de courbure les deux coniques focales d'un système de surfaces homofocales du second degré, Comptes Rendus Hebd. Séances Acad. Sciences 53, 799-801.
36. Sabin, M. (1989), Circles and cyclides - applying the absolute, in: Handscomb, D.C., ed., The Mathematics of Surfaces III, Clarendon Press, Oxford, 145-151.
37. Schmidt, H. (1950), Die Inversion und ihre Anwendungen, Oldenbourg, München.
38. Snyder, V. and Sisam, C.H. (1941), Analytic Geometry of Space. Henry Holt and Company, New York.
39. Ueda, K. (1995), Normalized cyclide Bézier patches, in: Daihlen, M., Lyche, T. and Sehumaker, L.L., eds., Mathematical Methods in CAGD III, Academic Press, Boston, to appear, 1-10.
40. Villarceau, Y. (1848), Extrait d'une note communiqué à M. Babinet, Comptes Rendus Hebd. Séances Acad. Sciences 27, 246.
41. Zhou, Z. and Strasser, W. (1991), A NURBS representation of cyclides, in: Kunii, T.L., ed., Modeling in Computer Graphics, Springer, Tokyo, 78-92.