Effective algorithms for the study of the degree of algebraic varieties in offsetting processes

Abstract

The research in this thesis is framed within the field of Symbolic Computation, and more specifically in the subfield of Effective (Symbolic) Algebraic Geometry of Curves and Surfaces. In particular, this thesis focuses on the study of the degree structure of the multivariate polynomial defining the geometric object generated when applying offsetting processes. That is, we study its total and partial degrees w.r.t. each variable, including the distance variable. In order to do this, the thesis is structured into four chapters and two appendixes, as follows. • Chapter 1 presents the notions of generic offset and generic offset polynomial, with their basic properties. This chapter provides the theoretical foundation of our subject of study. In particular, we prove the fundamental property of the generic offset polynomial; i.e., that this polynomial specializes to the polynomial defining the classic offset for all, but at most finitely many values of the distance. After establishing the connection with the classical theory, we define the central problem of this thesis: the degree problem for the generic offset. We also introduce the associated notation and terminology. The chapter includes also some technical lemmas about the use of resultants for the analysis of curve intersection problems. • Chapter 2 deals with the total degree problem for the generic offset of a plane curve. We consider the general case where the curve is given by its implicit equation and, for rational curves, we also consider the parametric representation of the curve. In both cases we provide efficient formulae for the total degree of the generic offset. Furthermore, we provide additional formulae that can be applied to obtain theoretical information about the total degree of the offset. In this chapter we will meet the notions of offset-line system, auxiliary curve and fake point. These three notions play an essential role in our approach to the degree problems studied in this thesis. We use them to develop a common framework for resultant-based degree formulae, that will be applied to several different degree problems in the following chapter. • Chapter 3 is a natural continuation of the preceding one. In this chapter we apply the strategy, methods and language of Chapter 2, to complete the analysis of the degree problem for plane curves. Thus, we provide efficient formulae for the partial degree and the degree w.r.t the distance variable of the generic offset, both in the implicit and parametric cases. Besides, we also provide formulae that are suitable for a theoretical analysis of these degree problems. • Chapter 4 deals with the degree problem for surfaces. The major part of this chapter is dedicated to present a total degree formula for rational surfaces, given parametrically. This formula can be applied under a very general assumption about the surface. Namely, we need to assume that there are at most finitely many distance values for which the offset of the surface passes through the origin (see Assumption 4.1, page 122). The formula requires the computation of a univariate generalized resultant and gcds, of polynomials with symbolic coefficients. In the final section of this chapter we apply an alternative approach, independent of the previous results in this chapter, to study the offset degree structure for surfaces of revolution. We provide a complete and efficient solution for this case. • Appendix A contains a summary of the degree formulae obtained in this thesis. Appendix B shows the results of some computations, corresponding to proofs or examples, that, due to their length, are more conveniently placed here.