Automatic parameterization of rational curves and surfaces IV: algebraic space curves

ABSTRACT

For an irreducible algebraic space curve C that is implicitly defined as the intersection of two algebraic surfaces, f (x, y, z) = 0 and g (x, y, z) = 0, there always exists a birational correspondence between the points of C and the points of an irreducible plane curve P, whose genus is the same as that of C. Thus C is rational if the genus of P is zero. Given an irreducible space curve C = (f ∩ g), with f and g not tangent along C, we present a method of obtaining a projected irreducible plane curve P together with birational maps between the points of P and C. Together with [4], this method yields an algorithm to compute the genus of C, and if the genus is zero, the rational parametric equations for C. As a biproduct, this method also yields the implicit and parametric equations of a rational surface S containing the space curve C. The birational mappings of implicitly defined space curves find numerous applications in geometric modeling and computer graphics since they provide an efficient way of manipulating curves in space by processing curves in the plane. Additionally, having rational surfaces containing C yields a simple way of generating related families of rational space curves.

REFERENCES

• 1: ABHYANKAR, ~. ~. Algebraic Space Curves. Les Presses de L'Universit~ de Montreal, Montreal, Canada, 1971.
• 2: S. S. Abhyankar , C. Bajaj, Automatic parameterization of rational curves and surfaces 1: conics and conicoids, Computer-Aided Design, v.19 n.1, p.11-14, Jan./Feb. 1987 [doi>10.1016/0010-4485(87)90147-3]
• 3: S. S. Abhyanker , C. Bajaj, Automatic parametrization of rational curves and surfaces II: cubics and cubicoids, Computer-Aided Design, v.19 n.9, p.499-502, Nov. 1987 [doi>10.1016/0010-4485(87)90235-1]
• 4: S. S. Abhyankar , C. L. Bajaj, Automatic parameterization of rational curves and surfaces III: algebraic plane curves, Computer Aided Geometric Design, v.5 n.4, p.309-321, Nov. 1988 [doi>10.1016/0167-8396(88)90011-8]
• 5: Chandrajit L. Bajaj, Geometric Modeling with Algebraic Surfaces, Proceedings of the 3rd IMA Conference on the Mathematics of Surfaces, p.3-48, September 19-21, 1988
• 6: BOCHER, M. Introducttion to Higher Algebra, MacMillan, New York, 1907.
• 7: 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]
• 8: W. S. Brown, The Subresultant PRS Algorithm, ACM Transactions on Mathematical Software (TOMS), v.4 n.3, p.237-249, Sept. 1978 [doi>10.1145/355791.355795]
• 9: BUCHBERGER, B. Grobner bases: An algorithmic method in polynomial ideal theory. In Recent Trends in Multidimensional System Theory, N. Bose, Ed. Reidel, 1984.
• 10: George E. Collins, Subresultants and Reduced Polynomial Remainder Sequences, Journal of the ACM (JACM), v.14 n.1, p.128-142, Jan. 1967 [doi>10.1145/321371.321381]
• 11: George E. Collins, The Calculation of Multivariate Polynomial Resultants, Journal of the ACM (JACM), v.18 n.4, p.515-532, Oct. 1971 [doi>10.1145/321662.321666]
• 12: Thomas Garrity , Joe Warren, On computing the intersection of a pair of algebraic surfaces, Computer Aided Geometric Design, v.6 n.2, p.137-153, May 1989 [doi>10.1016/0167-8396(89)90017-4]
• 13: HABICHT, W. Eine Verallgemeinrung des Sturmschen Wurzelzahlverfahrens. Comm. Math. T-L~I,,r,+;,~; 01 (1QAO~ QO_l 1
• 14: Loos, R. Generalized polynomial remainder sequences, in Computer Algebra, Symbolic and Algebraic Computation, 2d Ed., Buchberger, Collins, Loos, Albrecht, Eds. Wien, New York, 1983.
• 15: SALMON, G. f~sson-_s Introductory to t_he Modern Higher Algebra, Chelsea New York, 1885.
• 16: J. T. Schwartz, Fast Probabilistic Algorithms for Verification of Polynomial Identities, Journal of the ACM (JACM), v.27 n.4, p.701-717, Oct. 1980 [doi>10.1145/322217.322225]
• 17: T. W. Sederberg , J. P. Snively, Parametrization of cubic algebraic surfaces, Proceedings on Mathematics of surfaces II, p.299-319, October 1987, Cardiff, United Kingdom
• 18: SNYDER, V., AND SISAM, C. Analytic Geometry of Space, Henry Holt, New York, 1914.
• 19: WALKER, R. Algebraic Curves, Springer Verlag, New York, 1978.
• 20: ZARISKI, O., AND SAMUEL, P. Commutative Algebra, Vols. I and II. Springer Verlag, New York, 1958.

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:
  F. Theory of Computation
  F.2 ANALYSIS OF ALGORITHMS AND PROBLEM COMPLEXITY
      F.2.1 Numerical Algorithms and Problems
          Subjects: Computations on polynomials

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

  I. Computing Methodologies
  I.1 SYMBOLIC AND ALGEBRAIC MANIPULATION
  I.3 COMPUTER GRAPHICS
      I.3.5 Computational Geometry and Object Modeling
          Subjects: Geometric algorithms, languages, and systems

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

General Terms:
Algorithms, Design, Theory