The implicitization of a rational parameterisation σ: R^{2}→ R^{3} of the form:

(s,t) → (f_{1}(s,t)/f_{0}(s,t),f_{2}(s,t)/f_{0}(s,t), f_{3}(s,t)/f_{0}(s,t)),

where f_{0}, f_{1}, f_{2}, f_{3} are bivariate polynomials, consists in finding a polynomial P(X,Y,Z) of minimal degree, such that P(f_{1}(s,t)/f_{0}(s,t),f_{2}(s,t)/f_{0}(s,t), f_{3}(s,t)/f_{0}(s,t))=0, for all (s,t) such that f_{0}(s,t)≠0.

The minimal degree of the polynomial P, is the degree δ of the algebraic surface, which is the image of σ. It depends on the degree of the polynomial f_{i} and the existance of so-called base points. These are points (u,v) such that f_{0}(u,v)=…=f_{3}(u,v)=0, including such points at infinity. If the maximum degree of the polynomials f_{i} is d, then the algebraic degree

deg(σ).δ=d^{2} - ∑_{p base point}μ_{p,}

where μ_{p} is the (algebraic) multiplicity of the base point p and where deg(σ) denotes the (constant) generic number of distinct parameters that give the same point on the algebraic surface image of σ.

This operation, which corresponds to a change of representation, has important applications such as for instance, computing intersection curves or detecting singularities:

From an algebraic point of view, we consider the equations

f_{1}(s,t) - Xf_{0}(s,t)=0, f_{2}(s,t) - Yf_{0}(s,t)=0, f_{3}(s,t) - Zf_{0}(s,t)=0,

from which we want to deduce the implicit equation P(X,Y,Z)=0. The implicitization problem consists then of eliminating the variables (s,t) from this set of equations.

General resultant techniques, but also specialised methods have been reviewed or developed in the GAIA project to solve the implicitization process:

Published May 25, 2005

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