Foundations of Computational Mathematics

, Volume 10, Issue 1, pp 37–66 | Cite as

Linear Precision for Toric Surface Patches

  • Hans-Christian Graf von Bothmer
  • Kristian Ranestad
  • Frank Sottile
Open Access


We classify the homogeneous polynomials in three variables whose toric polar linear system defines a Cremona transformation. This classification includes, as a proper subset, the classification of toric surface patches from geometric modeling which have linear precision. Besides the well-known tensor product patches and Bézier triangles, we identify a family of toric patches with trapezoidal shape, each of which has linear precision. Furthermore, Bézier triangles and tensor product patches are special cases of trapezoidal patches.


Bézier patches Geometric modeling Linear precision Cremona transformation Toric patch 

Mathematics Subject Classification (2000)

14M25 65D17 


  1. 1.
    E. Casas-Alvero, Singularities of Plane Curves. London Mathematical Society Lecture Note Series, vol. 276 (Cambridge University Press, Cambridge, 2000). MATHGoogle Scholar
  2. 2.
    D. Cox, What is a toric variety? in Topics in Algebraic Geometry and Geometric Modeling. Contemporary Mathematics, vol. 334 (Amer. Math. Soc., Providence, 2003), pp. 203–223. Google Scholar
  3. 3.
    D. Cox, J. Little, D. O’Shea, Ideals, Varieties, and Algorithms, 3rd edn. Undergraduate Texts in Mathematics (Springer, New York, 2007). MATHGoogle Scholar
  4. 4.
    I.V. Dolgachev, Polar Cremona transformations, Mich. Math. J. 48, 191–202 (2000). Dedicated to William Fulton on the occasion of his 60th birthday. MATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    L. García-Puente, F. Sottile, Linear precision for parametric patches, Adv. Comput. Math. (2009, to appear). Google Scholar
  6. 6.
    R. Goldman, Pyramid Algorithms: A Dynamic Programming Approach to Curves and Surfaces for Geometric Modeling (Morgan Kaufmann/Academic Press, San Diego, 2002). Google Scholar
  7. 7.
    R. Hartshorne, Algebraic Geometry. Graduate Texts in Mathematics, vol. 52 (Springer, New York, 1977). MATHGoogle Scholar
  8. 8.
    K. Karčiauskas, R. Krasauskas, Comparison of different multisided patches using algebraic geometry, in Curve and Surface Design, ed. by P.-J. Laurent, P. Sablonniere, L.L. Schumaker. Saint-Malo, 1999 (Vanderbilt University Press, Nashville, 2000), pp. 163–172. Google Scholar
  9. 9.
    R. Krasauskas, Toric surface patches, Adv. Comput. Math. 17(1–2), 89–133 (2002). Advances in Geometrical Algorithms and Representations. MATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    R. Krasauskas, Bézier patches on almost toric surfaces, in Algebraic Geometry and Geometric Modeling. Mathematics and Visualization (Springer, Berlin, 2006), pp. 135–150. CrossRefGoogle Scholar
  11. 11.
    R. Krasauskas, Minimal rational parametrizations of canal surfaces, Computing 79(2–4), 281–290 (2007). MATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    H. Pottmann, J. Wallner, Computational Line Geometry. Mathematics and Visualization (Springer, Berlin, 2001). MATHGoogle Scholar
  13. 13.
    F. Sottile, Toric ideals, real toric varieties, and the moment map, in Topics in Algebraic Geometry and Geometric Modeling. Contemporary Mathematics, vol. 334 (Amer. Math. Soc., Providence, 2003), pp. 225–240. Google Scholar
  14. 14.
    B. Sturmfels, Solving Systems of Polynomial Equations. CBMS Regional Conference Series in Mathematics, vol. 97 (Amer. Math. Soc., Providence, 2002). MATHGoogle Scholar

Copyright information

© SFoCM 2009

Authors and Affiliations

  • Hans-Christian Graf von Bothmer
    • 1
  • Kristian Ranestad
    • 2
  • Frank Sottile
    • 3
  1. 1.Mathematisches InstitutGeorg-August-Universitiät GöttingenGöttingenGermany
  2. 2.Matematisk InstituttUniversitetet i OsloOsloNorway
  3. 3.Department of MathematicsTexas A&M UniversityCollege StationUSA

Personalised recommendations