Dupin cyclides

51 Matemáticas - Mathematics

Introduction: We will study the problem of finding a Dupin cyclide given three contact conditions. A contact condition is a point m in R3 and an orientation at that point, i.e., a vector with initial point m, this vector can be seen as the normal vector of an oriented plane through m. A Dupin cyclide that, historically, is defined as the envelope of a family of spheres which are simultaneously tangent to three given spheres, can be defined in the context of the projective five-dimensional space as 2-plane section of a quadric in this projective space. The points of this quadric correspond to oriented spheres in 3D.

Tipo de documento: Tesis-trabajos de grado - Thesis Maestría

Colaborador - Asesor: Paluszny Kluczynsky, Marco

Información adicional: Maestría en Ciencias - Matemáticas

Palabras clave: Dupin cyclides, Surfaces

Temática: 5 Ciencias naturales y matemáticas - Science 51 Matemáticas - Mathematics

Source: http://www.bdigital.unal.edu.co

Teaser

Dupin cyclides Alejandro Alberto Villa Isaza Thesis presented to obtain the title of: Master of Sciences - Mathematics Supervised by: Marco Paluszny Kluczynsky Universidad Nacional de Colombia - Medellı́n site Faculty Of Sciences, School of Mathematics 2014 To Martinga 2 Contents 1 Möbius Geometry 8 1.1 Projective Geometry . 1.2 The Möbius Space of Unoriented Spheres 10 1.3 General Cyclides .
15 2 Lie Sphere Geometry 8 19 2.1 The Space of Oriented Spheres 19 2.2 Λ4 and the Paraboloid Π .
22 3 Dupin Cyclides 3.1 26 Dupin Cyclides as 2-Plane Sections of Q4 .
27 4 Three Contact Conditions 30 4.1 Contact Conditions 30 4.2 The Homographies pang, pong and ping A Bézier Curves and Blossoms 33 41 A.1 Bézier Curves .
41 A.2 The De Casteljau Algorithm .
43 A.3 Blossoms 44 B Envelopes and Inversion in R3 46 B.1 Envelopes .
46 B.2 Inversion in R3 47 3 List of Figures 1.1 Projective Space P4 1.2 Stereographic Projection 11 1.3 Pencils of Spheres .
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