Topology of families of implicit algebraic surfaces depending on a parameter

Authors:
Juan Gerardo Alcázar
Affiliations:
Departamento de Matemáticas, Universidad de Alcalá, Madrid, Spain
Venue:
CASC'11 Proceedings of the 13th international conference on Computer algebra in scientific computing
Year:
2011

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Abstract

Given a family of algebraic surfaces, implicitly defined, depending on a parameter t, here we provide an algorithm for computing the different shapes arising in the family. The algorithm decomposes the real line into finitely many pieces (points and intervals) so that over each interval the shape is invariant, in the sense that the topology of the family can be described by means of the same simplicial complex. As a consequence, by applying known algorithms ([1], [6], [7], [11]) the different shapes in the family can be computed. The algorithm is due to a generalization of the ideas in [2] to the surface case.