Foundations of Computational Mathematics

, Volume 10, Issue 1, pp 37-66

First online:

Open Access This content is freely available online to anyone, anywhere at any time.

Linear Precision for Toric Surface Patches

  • Hans-Christian Graf von BothmerAffiliated withMathematisches Institut, Georg-August-Universitiät Göttingen
  • , Kristian RanestadAffiliated withMatematisk Institutt, Universitetet i Oslo
  • , Frank SottileAffiliated withDepartment of Mathematics, Texas A&M University Email author 


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