Foundations of Computational Mathematics

, Volume 10, Issue 1, pp 37–66

Linear Precision for Toric Surface Patches


  • Hans-Christian Graf von Bothmer
    • Mathematisches InstitutGeorg-August-Universitiät Göttingen
  • Kristian Ranestad
    • Matematisk InstituttUniversitetet i Oslo
    • Department of MathematicsTexas A&M University
Open AccessArticle

DOI: 10.1007/s10208-009-9052-6

Cite this article as:
Graf von Bothmer, H., Ranestad, K. & Sottile, F. Found Comput Math (2010) 10: 37. doi:10.1007/s10208-009-9052-6


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

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© SFoCM 2009