A multisided generalization of Bézier surfaces

ABSTRACT

In this paper we introduce a class of surface patch representations, called S-patches, that unify and generalize triangular and tensor product Bézier surfaces by allowing patches to be defined over any convex polygonal domain; hence, S-patches may have any number of boundary curves. Other properties of S-patches are geometrically meaningful control points, separate control over positions and derivatives along boundary curves, and a geometric construction algorithm based on de Casteljau's algorithm. Of special interest are the regular S-patches, that is, S-patches defined on regular domain polygons. Also presented is an algorithm for smoothly joining together these surfaces with Ck continuity.

REFERENCES

• 1: Wolfgang Böhm , Gerald Farin , Jürgen Kahmann, A survey of curve and surface methods in CAGD, Computer Aided Geometric Design, v.1 n.1, p.1-60, July 1984 [doi>10.1016/0167-8396(84)90003-7]
• 2: design. Comput. Aided Geom. Des. 1, 1 (July 1984), 87-94.
• 3: Hiroaki Chiyokura , Fumihiko Kimura, Design of solids with free-form surfaces, ACM SIGGRAPH Computer Graphics, v.17 n.3, p.289-298, July 1983
• 4: DE BOOR, C. B-form basics. In Geometric Modeling." Algorithms and New Trends, G. Farin, Ed. SIAM, Philadelphia, Pa., 1987, pp. 131-148.
• 5: Anthony David Derose, Geometric continuity: a parametrization independent measure of continuity for computer aided geometric design (curves, surfaces, splines), 1985
• 6: Tony D. DeRose, Composing Bézier simplexes, ACM Transactions on Graphics (TOG), v.7 n.3, p.198-221, July 1988 [doi>10.1145/44479.44482]
• 7: DEROSE, T.D. Geometric programming: A coordinate-free approach. In SIGGRAPH 88 Tutorial Course Notes, Course No. 25, ACM, New York, Aug. 1988.
• 8: DEROSE, T. D., GOLDMAN, R. U., HAGEN, H., AND MANN, S. Composing rational B-splines. In preparation.
• 9: DODSON, C. T. J., AND POSTON, T. Tensor Geometry: The Geometric Viewpoint and Its Uses. Paperback ed. Pitman, London, 1979.
• 10: FARIN, G. A construction for visual C~ continuity of polynomial surface patches. Comput. Graph. image Process. 20, 3 (Nov. i982), 272-282.
• 11: FARIN, G. Smooth interpolation to scattered 3D data. In Surfaces in CAGD, R. E. Barnhill and W. Boehm, Eds. North-Holland, Amsterdam, 1983, pp. 43-63.
• 12: G Farin, Triangular Berstein-Be´zier patches, Computer Aided Geometric Design, v.3 n.2, p.83-127, August 1986 [doi>10.1016/0167-8396(86)90016-6]
• 13: Gerald Farin, Curves and surfaces for computer aided geometric design: a practical guide, Academic Press Professional, Inc., San Diego, CA, 1988
• 14: GREGORY, J.A. Cl rectangular and non-rectangular surface patches. In Surfaces and CAGD, R. E. Barnhill and W. Boehm, Eds. North-Holland, Amsterdam, 1983, pp. 25-33.
• 15: GREC, ORY, J.A. N-sided surface patches. In Mathematics of Surfaces, J. Gregory, Ed. Clarendon Press, Oxford~ E_ng!and, 1986, ppo 217-232o
• 16: John Gregory , Jörg Hahn, Geometric continuity and convex combination patches, Computer Aided Geometric Design, v.4 n.1-2, p.79-89, July 1987 [doi>10.1016/0167-8396(87)90026-4]
• 17: HERRON, G.J. Triangular and multisided patch schemes. Ph.D. thesis, Dept. of Mathematics, Univ. of Utah, Salt Lake City, 1979.
• 18: HERRON, G.J. Techniques for visual continuity. In Geometric Modeling." Algorithms and New Trends, G. Farin, Ed. SIAM, Philadelphia, Pa., 1987, pp.163-174.
• 19: HOSAKA, M., AND KIMURA, F. Non-four-sided patches expressions with control points. Comput. Aided Geom. Des. i, 1 (july 1984), 75-86.
• 20: JENSEN, T. Assembling triangular and rectangular patches and multivariate splines. In Geometric Modeling: Algorithm,~ and New Trends, G. Farin, Ed. SIAM, Philadelphia, Pa., 1987, pp. 203-22O.
• 21: PIPER, B. R. Visually smooth interpolation with triangular B~zier patches. In Geometric Modeling: Algorithms and New Trends, G. Farin, Ed. SIAM, Philadelphia, Pa., 1987, pp. 221- 233.
• 22: RAMSHAW, L. Blossoming: A connect-the-dots approach to splines. Res. Rep. 19, Digital Equipment Corp Systems Research Center, Palo Alto, Calif., June 21, 1987.
• 23: RAMSHAW, L. B~ziers and B-splines as multiaffine maps. In Theoretical Foundations of Computer Graphics and CAD, R. A. Earnshaw, Ed. NATO ASI Series, vol. F40, Springer-Verlag, New York, 1988, pp. 757-776.
• 24: RAMSHAW, L. Blossoms are polar forms. Res. Rep. 34, Digital Equipment Corp Systems Research Center, Palo Alto, Calif., Jan. 1989.
• 25: SAB~N, M.A. Non-rectangular surface patches suitable for inclusion in a B-spiine surface, in Proceedings of Eurographics '83, P. ten Hagen, Ed. North-Holland, Amsterdam, 1983, pp. 57-69.
• 26: SHIRMAN, L. A., AND SI~QUIN, C.H. Local surface interpolation with BSzier patches. Comput. Aided " "~
• 27: T. Varady, Survey and new results in n-sided patch generation, Proceedings on Mathematics of surfaces II, p.203-235, October 1987, Cardiff, United Kingdom

INDEX TERMS

Primary Classification:
  I. Computing Methodologies
  I.3 COMPUTER GRAPHICS
      I.3.5 Computational Geometry and Object Modeling
          Subjects: Curve, surface, solid, and object representations

Additional Classification:
  J. Computer Applications

General Terms:
Algorithms, Design, Theory