Blending parametric surfaces

ABSTRACT

A blending surface is a surface that smoothly connects two given surfaces along two arbitrary curves, one on each surface. This is particularly useful in the modeling operations of filleting a sharp edge between joining surfaces or connecting disjoint surfaces. In this paper we derive a new surface formulation for representing surfaces which are blends of parametric surfaces. The formulation has the advantage over the traditional rational polynomial approach in that point and normal values have no gaps between the blending surface and the base surfaces. Shape control parameters that control the cross-sectional shape of the blending surface are also available. In addition, the base surfaces are not restricted to any particular type of surface representation as long as they are parametrically defined and have a well-defined and continuous normal vector at each point. The scheme is extensible to higher orders of geometric continuity, although we concentrate on G1.

REFERENCES

• 1: Anthony David Derose, Geometric continuity: a parametrization independent measure of continuity for computer aided geometric design (curves, surfaces, splines), 1985
• 2: FAmN, G.E. A construction for visual C1 continuity of polynomial surface patches. Comput. Graph. Image Process. 20 (1982), 272-282.
• 3: FARIN, G. E., Ed. Geomet,,ic Modeling: Algorithms and New Trends. SIAM, Philadelphia, Pa., 1987.
• 4: Daniel J Filip, Adaptive subdivision algorithms for a set of Bezier triangles, Computer-Aided Design, v.18 n.2, p.74-78, March 1986 [doi>10.1016/0010-4485(86)90153-3]
• 5: FIuP, D. J., AND BALL, T.W. Procedurally ~epresenting lofted surfaces. IEEE Comput. Graph. Appl. To be published.
• 6: Daniel Filip , Robert Magedson , Robert Markot, Surface algorithms using bounds on derivatives, Computer Aided Geometric Design, v.3 n.4, p.295-311, 1987 [doi>10.1016/0167-8396(86)90005-1]
• 7: I. D. Faux , M. J. Pratt, Computational Geometry for Design and Manufacture, Halsted Press, New York, NY, 1979
• 8: GORDON, W.J. Spline-blended surface interpolation through curve networks. J. Math. and Mech. 18, 10 (1969), 931-957.
• 9: Christoph Hoffmann , John Hopcrott, Quadratic blending surfaces, Computer-Aided Design, v.18 n.6, p.301-306, July/Aug. 1986 [doi>10.1016/0010-4485(86)90091-6]
• 10: LANE, J. M., AND RIESENFFLD, R.F. A theoretical development for the computer generation and display of piecewise polynomial surfaces. IEEE Trans. Pattern Anal. Machine Intell. 2, 1 (1980), 35-46.
• 11: ROCKWOOB, A.P. Introducing sculptured surfaces into a geometric modeler. In Solid Modeling by Computers: From Theory to Application, M. S. Pickett and J. W. Boyce, Eds., Plenum Press, New York, 1984, pp. 237-253.
• 12: ROSSI~NAC, J. R., AND REQUICHA, A.G. A constant-radius blending in solid modeling. Comput. Mech. Eng. (July 1984), 65-73.
• 13: Joe David Warren, On algebraic surfaces meeting with geometric continuity, Cornell University, Ithaca, NY, 1986

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:
  G. Mathematics of Computing
  G.1 NUMERICAL ANALYSIS
      G.1.1 Interpolation
          Subjects: Spline and piecewise polynomial interpolation

General Terms:
Algorithms, Theory