Metodología proyectiva en espacios no convencionales: Aplicación a variedades unicursales y curvas de diseño

Abstract

The first chapter is devoted to set out the working plan; it justifies the content and outlines novel and applied aspects of this work. Chapter II includes the basic concepts behind the entire dissertation. The abstract concept of projective space is briefly recalled here and all those spaces considered as conventional spaces that have been widely analyzed by classical authors are also mentioned in this chapter. Below, some of them whose study had scarcely been approached by the structuralist principles of the Algebra have been listed within the graphical environment. This chapter ends with the introduction of what we consider is a new concept. We are speaking of spaces of bipoints, birays, tripoints, etc. Those spaces will greatly assist us in deducing certain graphical properties. Chapter III is entirely devoted to the generalization, in all its forms, of the concept of involution, there we establish results as curious as the fact that the involutions of the points of a line are the lines of the two-dimensional space of bipoints contained in it. Once the foundations of the work are established, we move into the Chapter IV to develop the main concept of this dissertation: unicursal varieties. The general concept of unicursal variety is defined at that point, and then we focus immediately on the explanation of the very unicursal varieties. We pushed aside a huge number of useful spaces such as ruled surfaces with unicursal cuspidal edge, polar surfaces of unicursal curves, twisted cubics, unicursal twisted quartics, etc. Any of these topics duly developed could be the subject of a new thesis. In this chapter a general theorem about involutions is discussed, it widespread generalize theorems as classic as the Descargues’theorem applied to families of conics. Some examples of certain complexity are shown in this chapter, where the power of the theorem can be appreciated for deducing complex graphical properties through scant calculations. At some stage, we resort to use symbolic computation in order to save some tedious steps...