q(x,y) = ∑i+j≤m c
q(x,y,z) = ∑
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:
Published June 29, 2005