Simplification of surface parametrizations

ABSTRACT

Given a rational parametrization of an algebraic surface, we try to reduce the degree by a suitable reparametrization. We give an algorithm that produces a parametrization with a degree that is at most twice the minimal degree. The problem is closely related to the simplification of linear systems of plane curves by Cremona transformations.

REFERENCES

• 1: Carlos Andradas , Tomás Recio , J. Rafael Sendra, A relatively optimal rational space curve reparametrization algorithm through canonical divisors, Proceedings of the 1997 international symposium on Symbolic and algebraic computation, p.349-355, July 21-23, 1997, Kihei, Maui, Hawaii, United States [doi>10.1145/258726.258844]
• 2: Carlos Andradas , Taomás Recio , J. Refael Sendra, Base field restriction techniques for parametric curves, Proceedings of the 1999 international symposium on Symbolic and algebraic computation, p.17-22, July 28-31, 1999, Vancouver, British Columbia, Canada [doi>10.1145/309831.309845]
• 3: G. Castelnuovo. Ricerche generali sopra sui sistemi lineari di curve piane. In Memorie scelte, pages 137-187. Zanichelli, 1891.
• 4: F. Conforto. Le superfici razionali. Zanichelli, 1939.
• 5: Jamie Gutierrez , Rosario Rubio , David Sevilla, Unirational fields of transcendence degree one and functional decomposition, Proceedings of the 2001 international symposium on Symbolic and algebraic computation, p.167-174, July 2001, London, Ontario, Canada [doi>10.1145/384101.384124]
• 6: Hoon Hong , Josef Schicho, Algorithms for trigonometric curves (simplification, implicitization, parameterization), Journal of Symbolic Computation, v.26 n.3, p.279-300, Sept. 1998 [doi>10.1006/jsco.1998.0212]
• 7: Y. Manin. Cubic Forms. North-Holland, 1974.
• 8: M. Nagata. Rational surfaces I + II. Mem. Coll. Sci. Kyoto, 32 and 33:351-370+271-293, 1960.
• 9: J. Schicho. Rational parametrization of surfaces. PhD thesis, RISC Linz, 1995.
• 10: J. Schicho. Inversion of rational maps with Gröbner bases. In B. Buchberger and F. Winkler, editors, Gröbner bases and applications, pages 495-503. Cambridge Univ. Press, 1998.
• 11: Josef Schicho, Rational parametrization of real algebraic surfaces, Proceedings of the 1998 international symposium on Symbolic and algebraic computation, p.302-308, August 13-15, 1998, Rostock, Germany [doi>10.1145/281508.281655]
• 12: Josef Schicho, Rational parametrization of surfaces, Journal of Symbolic Computation, v.26 n.1, p.1-29, July 1998 [doi>10.1006/jsco.1997.0199]
• 13: J. Schicho. A degree bound for the parameterization of a rational surface. J. Pure Appl. Alg., 145:91-105, 1999.
• 14: T W Sederberg, Improperly parametrized rational curves, Computer Aided Geometric Design, v.3 n.1, p.67-75, May 1986 [doi>10.1016/0167-8396(86)90025-7]
• 15: J. Rafael Sendra , Franz Winkler, Computation of the degree of rational maps between curves, Proceedings of the 2001 international symposium on Symbolic and algebraic computation, p.317-322, July 2001, London, Ontario, Canada [doi>10.1145/384101.384144]
• 16: I. R. Shafarevich, editor. Algebraic surfaces. Proc. Steklov Inst. Math., 1965. transl. by AMS 1967.
• 17: Mark van Hoeij, Rational parametrizations of algebraic curves using a canonical divisor, Journal of Symbolic Computation, v.23 n.2-3, p.209-227, Feb./March 1997 [doi>10.1006/jsco.1996.0084]

INDEX TERMS

Primary Classification:
  I. Computing Methodologies
  I.1 SYMBOLIC AND ALGEBRAIC MANIPULATION
      I.1.2 Algorithms
          Subjects: Analysis of algorithms

Additional Classification:
  G. Mathematics of Computing
  G.1 NUMERICAL ANALYSIS
      G.1.3 Numerical Linear Algebra
          Subjects: Linear systems (direct and iterative methods)
      G.1.5 Roots of Nonlinear Equations
          Subjects: Polynomials, methods for

General Terms:
Algorithms, Standardization, Theory