Page 20 - Geometric Modeling and Algebraic Geometry

14 T. Dokken
We have collected known results about such classifications, especially concern-
ing results for real curves and surfaces of low degree. Of particular interest in CAGD
are parameterizable (i.e. so-called rational) curves and surfaces, and we have made
explicit studies of various such objects. These objects, or patches of these objects, are
potential candidates for approximate implicitization problems. For example, when
the rough shape of a patch to be approximated is known, one can choose from a
“catalogue” what kind of parameterized patch that is suitable - this eliminates many
unknowns in the process of finding an equation for the approximating object and
will therefore speed up the application. In addition to the survey of known results,
particular objects that have been studied are:
• monoid curves and surfaces, especially quartic monoid surfaces
• tangent developables
• triangle and tensor surfaces of low degree of low (bi)degrees
Papers from the project addressing classification are [41, 42].

1.6 Exact and approximate implicitization

In CAD-type algorithms, combining parametric and algebraic representation of sur-
faces is in many algorithms advantageous. However, for surfaces of algebraic degree
higher than two this is in general a very challenging task. E.g., a rational bi-cubic
surface has algebraic degree 18. All rational surfaces have an algebraic representa-
tion. However, for surfaces of total degree higher than 3, not all algebraic surfaces
will have a rational parametric representation. In the project we have the following
two main approaches for change of representation.

1.6.1 Exact implicitization of rational parametric surfaces

General resultant techniques, but also specialized methods have been reviewed or
developed in the GAIA II project to address the implicitization process:
• Projective, as well as anisotropic, resultants when the polynomials f 0 ,...,f 3
have no base points.
• Residual resultants when the polynomials have 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 the polynomials as soon as the base points are locally
defined by at most two equations.
Papers from the project addressing topics of exact implicitization are [6, 23, 24, 27,
47].