Blending quadric surfaces with quadric and cubic surfaces

ABSTRACT

We show that a pair of quadric surfaces, M and N, may be blended (smoothly joined) using a quadric surface if and only if the pencil of M and N contains either a plane or a pair of planes. Under this condition, we derive the equations of quadric and cubic surfaces that blend M and N.

REFERENCES

• 1: A. Dresden, Solid Analytic Geometry and Determinants, Dover, New York, (1964).
• 2: Joshua Levin, A parametric algorithm for drawing pictures of solid objects composed of quadric surfaces, Communications of the ACM, v.19 n.10, p.555-563, Oct. 1976 [doi>10.1145/360349.360355]
• 3: C. Hoffmann and J. Hopcroft, "Automatic Surface Generation in Computer Aided Design," The Visual Computer, Vol. 1, No. 2, Springer- Verlag, 1985, pp. 92-100.
• 4: Christoph M. Hoffmann , John E. Hopcroft, Quadratic Blending Surfaces, Cornell University, Ithaca, NY, 1985
• 5: Christoph M. Hoffmann , John E. Hopcroft, The Potential Method for Blending Surfaces and Corners, Cornell University, Ithaca, NY, 1985
• 6: Alan E. Middleditch , Kenneth H. Sears, Blend surfaces for set theoretic volume modelling systems, ACM SIGGRAPH Computer Graphics, v.19 n.3, p.161-170, Jul. 1985
• 7: A. ItO~~liW~~Otl id J. Owm, "l~lc:l~tlil~g ,Surribcc:s in Solid Cc:ollrc:l,ric Moclclillg," S'IA M ChtJ. on CYcornelric Modeling Ural Roboks, A I hny NY (1985).
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INDEX TERMS

Primary Classification:
  I. Computing Methodologies
  I.3 COMPUTER GRAPHICS
      I.3.7 Three-Dimensional Graphics and Realism
          Subjects: Visible line/surface algorithms

Additional Classification:
  F. Theory of Computation
  F.2 ANALYSIS OF ALGORITHMS AND PROBLEM COMPLEXITY
      F.2.2 Nonnumerical Algorithms and Problems
          Subjects: Geometrical problems and computations

  G. Mathematics of Computing
  G.1 NUMERICAL ANALYSIS
      G.1.1 Interpolation
          Subjects: Smoothing

  I. Computing Methodologies
  I.3 COMPUTER GRAPHICS
      I.3.5 Computational Geometry and Object Modeling
          Subjects: Curve, surface, solid, and object representations
  I.4 IMAGE PROCESSING AND COMPUTER VISION
      I.4.3 Enhancement
          Subjects: Smoothing

  J. Computer Applications
  J.6 COMPUTER-AIDED ENGINEERING
      Subjects: Computer-aided design (CAD)

General Terms:
Algorithms, Theory