Introduction to Symbolic-Numeric Computational Algebra (Lecture 3)
Dealing with approximate geometric problems with a symbolic-numeric approach

Laureano Gonzalez-Vega (University of Cantabria)

This lecture and the next will be devoted to show how symbolic techniques such as Grobner Bases, Multivariate Resultants, Cylindrical Algebraic Decomposition, etc can be used in conjunction with methods from Numerical Linear Algebra such as the singular value decomposition or semidefinite programming in order to solve in a more accurate and reliable way intersection problems for (real) curves and surfaces in Computer Aided Geometric Design.

Central problems to be considered will be approximate implicitation, topology computation for implicitly (and approximately) defined curves and surfaces and offset sectioning.

Two different situations will be highlighted:

  • symbolic techniques (specially adapted) are used to prepare the appropiate application of some numerical algorithm such as computing the singular values of a matrix;
  • numerical techniques are used to compute, for example, a good implicit approximation of a given rational surface in order to apply later symbolic algorithms to get topological/shape information about the considered surface.