Implicitization

Implicitization is the process of finding implicit equations for a curve or surface specified by parametric equations. For parameterizations by polynomials or rational functions over , the algebraic problem is to find polynomial equations defining the smallest affine variety containing the image. This variety can contain points that the original parameterization does not reach (Abbott et al. 2017).

For a polynomial parameterization over a field , form the ideal generated by in . Its elimination ideal in consists precisely of the polynomials that vanish identically after substituting the parameterization. A Gröbner basis for a lexicographic order placing all parameters before all coordinate variables computes this ideal.

For example, eliminating the parameter from

			(1)
			(2)

where , gives the implicit equation

(3)

The parametric equations cover the unit circle except for , which is recovered as a limit as . Thus the implicit equation includes a point missing from the original image.

The figure above illustrates how clearing denominators without retaining their nonvanishing conditions can introduce additional points. For example, the parameterization , with has , but the cleared equations and give

(4)

The resultant vanishes on the entire line as well as on the intended line . The Rabinowitsch trick introduces an auxiliary variable with to enforce . Eliminating computes an ideal saturation, and then eliminating leaves as the implicit equation. The point still belongs to the resulting line even though it is absent from the parameterization.

Standard algebraic methods use resultants or Gröbner bases (Manocha and Canny 1992, D'Andrea 2001, Abbott et al. 2017). For a canal surface, the equations for the moving sphere and its envelope provide a starting point for eliminating the parameter, as in the construction of a tube around a trefoil knot (Trott 2006). Restrictions on the ranges of real parameters must also be checked separately.