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Algebraic curves and surfaces
An algebraic curve  of degree m>0 is described by the zero set of a polynomial q(x,y)=0 of total degree m. An algebraic surface of degree m>0 is described by the zero set of a polynomial q(x,y,z)=0 of total degree m.
  • q(x,y) = ∑i+j≤m c

  • q(x,y,z) = ∑

The algebraic degree of parametric surfaces

From algebraic geometry it is well known that a rational parametric curve of degree n has an algebraic representation of total degree n. For rational parametric tensor product surfaces of degrees (n1, n) the algebraic representation is an algebraic surface of total degree 2n1n. Consequently a bi-cubic NURBS surface, which is a typical CAD-type surface, has an algebraic degree of 18.


For algebraic curves and surfaces of total degree 1 and total degree 2 there is always a rational parameterization. However, for algebraic degrees higher than 2 this is often not a case.



The process of finding the algebraic description of a parametric curve or surface is called implicitization. Two main approaches exist:

  • Within exact implicitization, based on exact arithmetic, a number of classical and newer methods exist. Such methods are much too slower for use in typical industrial applications.
  • Approximate implicitization, is an approach that finds an implicit approximation within specified tolerances to a parametric curves, surface or higher dimensional manifold. The approach of approximate implicitization is aimed at the requirements of industrial applications. Although the approach is denoted "approximate implicitization" it will work as an exact implicitization method if exact arithmetic and a correct algebraic degree are employed.

Published June 29, 2005