Page 18 - Geometric Modeling and Algebraic Geometry

12 T. Dokken
q 2 (x, y, z)=0 of p 2 (s, t). Combining the parametric description of p 1 (u, v) and
the implicit representation q 2 (x, y, z)=0 of p 2 (s, t), we get q 2 (p 1 (u, v)) = 0. This
is a tensor product polynomial of bi-degree (54, 54). The intersection of two bicubic
patches is converted to finding the zero of

54 54

i j
q 2 (p 1 (u, v)) = c i,j u v =0.
i=0j=0
This polynomial has 55 × 55 = 3025 monomials with corresponding coefficients,
describing an algebraic curve of total degree 108. This illustrates that the intersection
of seemingly simple surfaces can results in a very complex intersection topology. As
in the case of curves, the surfaces we consider in CAGD are bounded, and we are
interested in the solution only in a limited interval (u, v) ∈ [a, b] × [c, d].

1.5 Extend the use of algebraic geometry within CAD

The work within the GAIA project related to algebraic geometry and CAD has ad-
dressed three main topics:

• Resultants are one of the traditional methods for exact implicitization of rational
parametric curves and surfaces. GAIA has produced some new results within this
classical research area.
• Singularities in algebraic curves and surfaces are for understanding their geom-
etry and topology.
• Classification is an old tradition in the field of Algebraic Geometry. It is a natural
starting point when trying to understand the geometry of algebraic objects.

Papers on CAGD and algebraic methods from the project are [8, 9, 32, 33, 34, 35,
41, 42, 44, 48, 49, 57].

1.5.1 Resultants
The objective has been to develop tools for constructing, manipulating and exploit-
ing implicit representations for parametric curves and surfaces based on resultant
computations. The work in GAIA has been divided into three parts:

• A survey in four parts addressing:
1. A resultant approach to detecting intersecting curves in P .
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2. Implicitizing rational hypersurfaces using approximation complexes.
3. Using projection operators in Computer Aided Design.
4. The method of moving surfaces for the implicitization of rational parametric
surface in P .
3
• A report addressing sparse/toric resultant, results when the number of monomials
is small compared to the number of possible monomials for polynomial of the
degree in question.