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Exact implicitization

The implicitization of a rational parameterisation σ: R2R3 of the form:

                     (s,t) →  (f1(s,t)/f0(s,t),f2(s,t)/f0(s,t), f3(s,t)/f0(s,t)),

where f0, f1, f2, f3 are bivariate polynomials, consists in finding a polynomial P(X,Y,Z) of minimal degree, such that P(f1(s,t)/f0(s,t),f2(s,t)/f0(s,t), f3(s,t)/f0(s,t))=0, for all (s,t) such that f0(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 fi and the existance of so-called base points. These are points (u,v) such that f0(u,v)=…=f3(u,v)=0, including such points at infinity. If the maximum degree of the polynomials fi is d, then the algebraic degree

                     deg(σ).δ=d2 - ∑p base pointμp,

where μp 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:

  • By susbtituting the parameterisation of a surface into the implicit equation of the other, we directly get the implicit equation of the intersection curve.
  • By computing the singular points from the implicit equation of a parameterised surface, one also obtains the self-intersection points of the surface.

From an algebraic point of view, we consider the equations

                     f1(s,t) - Xf0(s,t)=0, f2(s,t) - Yf0(s,t)=0, f3(s,t) - Zf0(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:

  • Projective, as well as anisotropic, resultants when the polynomials f0 ,…, f3 have no base point.
  • Residual resultants when σ has base points which are known and have special properties.
  • Determinants of the so-called approximation complexes which give an implicit equation of the image of σ as soon as the base points are locally defined by at most two equations.

Published May 25, 2005