Estudio asintótico de curvas y superficies

Abstract

This memory framework is the area of algebraic geometry, symbolic calculus and its applications in Computer Aided Geometric Design (CAGD). Its aim is to advance, from the preliminary studies related to the characterization of plane algebraic curves, towards the analysis of the branches’ properties of a curve at points with “sufficiently large” coordinates and towards the construction of the “generalized asymptotes” of a curve, extending the investigation to the case of algebraic surfaces. New symbolic computation methods are created, which characterize plane algebraic curves and analyze the behavior of their infinite branches, being able to extract a large part of the information about the behavior of a curve, study the topology of certain algebraic varieties, plane curves and surfaces in three dimensions, and represent them graphically at infinity. In addition, algorithms and implementations that effectively construct the generalized asymptotes of a given curve are presented, along with an analysis of their performance. It is, therefore, an investigation that involves two scientific disciplines: mathematics and computing, making mathematical expressions and mathematical objects can be handled, using symbolic calculus, to solve real-world problems. Thus, the main contributions and innovations of this doctoral thesis are: (1) the development of effective and exact methods for the construction of infinite branches and asymptotes of a plane algebraic curve, by calculating limits and derivatives, (2) the design and implementation of efficient algorithms for the calculation of the asymptotes of plane algebraic curves (with the algebra software Maple), as well as the analysis of their computational performance, (3) the determination of certain properties obtained from infinite branches and the construction of families of curves from certain asymptotes and (4) the definition of concepts of infinite branch, convergent infinite and approaching branches, applied to algebraic surfaces. These results can be adapted to the n-dimensional space and to curves defined by not necessarily rational parameterizations. In addition, future lines of applied research are opened in the 3-D graphical design area, in the field of data engineering, or in the field of performance analysis and computational efficiency of algorithms, among other areas of research and developing.