Superficies canales

Abstract

En este Trabajo Fin de Grado estudiamos las Superficies Canales, un tipo de superficie muy utilizada actualmente en el diseño asistido por ordenador por tener parametrizaciones racionales, aunque su definición y estudio original se debe a geómetras tan conocidos como Monge, Dupin o físicos como Maxwell. Para ello, primero recordamos una serie de conceptos básicos de Geometría Diferencial. A continuación, abordamos el concepto de los Conjuntos Focales, que nos va a ser necesario para poder caracterizar las Superficies Canales. Para finalizar, describimos las tres familias más representativas de este tipo de superficies: las tubulares, las de revolución y las cíclides de Dupin. Todo el texto está acompañado de numerosas ilustraciones que ayudan al lector a entender mejor los conceptos, haciendo de este TFG un trabajo visualmente más intuitivo.

This document shows the development of the Bachelor’s Degree Final Project, including all the acquired knowledge and the way in which its creator has worked in order to meet the objectives. Where possible, illustrations have been attached to the explanations and definitions, since it will make them more understandable and intuitive. Briefly, the purpose of this project is to perform a study of Canal Surfaces, a very popular kind of surface in computer-aided design currently, since they have rational parameterizations. However, it is important to take into account that their definition and original study were carried out by well-known geometers as Monge or Dupin, and physicists like Maxwell. First, a number of basic concepts of differential geometry will be collected. Once this task is completed, we will focus on one particular concept: focal sets, which are required to be able to characterize Canal Surfaces. And last, but not least, we will describe the three most representative types of this kind of surfaces: tubular surfaces, Dupin Cyclides and surfaces of revolution.