Portada

Agradecimientos y Dedicatorias

Resumen

Capítulo 1.  Introduction

Capítulo 2.  Theoretical frame and previous work

  • 2.1 Terminology
  • 2.2 Schemes for the modeling of n-Dimensional Polytopes
  • 2.3 Topological properties of 4D Orthogonal Pseudo-Polytopes
  • 2.4 Conclusions

Capítulo 3.  Configurations in the n-Dimensional Orthogonal Pseudo-Polytopes

  • 3.1 Introduction
  • 3.2 Configurations for the nD-OPP´s and the equivalence relation Rf
  • 3.3 The problem of determining the configurations for the nD-OPP´s (n > 4)
  • 3.4 Binary representation for the configurations in the nD-OPP´s
  • 3.5 Pólya´s countings and the number of configurations for the nD-OPP´s
  • 3.6 Four equivalence relations in the set Bn
  • 3.7 Fast comparison of combinations of nD hyper-boxes through relations Radj, RH & RE
  • 3.8 The ´Test-Box´ algorithm
  • 3.9 Conclusions

Capítulo 4.  The odd edge characterization and its role in the combinatorial topology of the n-Dimensional Orthogonal Pseudo-Polytopes

  • 4.1 Local analysis over the nD-OPP´s
  • 4.2 Spivak´s k-chains fundamental Concepts
  • 4.3 Linking k-chains with Topological Local Analysis of Odd Edges
  • 4.4 Conclusions

Capítulo 5.  Orthogonal polytopes modeling through the Extreme Vertices Model in the n-Dimensional Space (nD-EVM)

  • 5.1 Preliminary background
  • 5.2 Brinks and extreme vertices in the nD-OPP´s
  • 5.3 Local analysis over extreme vertices in the nD-OPP´s
  • 5.4 Global analysis over the nD-OPP´s and the extreme vertices model
  • 5.5 Relating sections and couplets
  • 5.6 Regularized boolean operations on the nD-EVM
  • 5.7 nD-EVM properties
  • 5.8 Conclusions

Capítulo 6.  Algorithms in the nD-EVM and their performance

  • 6.1 Basic algorithms for the nD-EVM
  • 6.2 The boolean operations algorithm for the nD-EVM
  • 6.3 Computing the content of an nD-OPP
  • 6.4 Computing the content of the boundary of an nD-OPP
  • 6.5 Computing forward and backward differences of an nD-OPP
  • 6.6 Algorithms for converting the nD-EVM to and from other schemes
  • 6.7 Conclusions

Capítulo 7.  Applications

  • 7.1 Application 1: Representing color 2D-animations through 4D-OPP´s and the EVM
  • 7.2 Application 2: A procedure for comparing color 2-dimensional images through their extrusions to the 5-dimensional colorspace
  • 7.3 Application 3: Incorporating the nD-EVM to image based reasoning
  • 7.4 Application 4: Manipulating "Real World" 3D datasets with the nD-EVM
  • 7.5 Application 5: Collision detection
  • 7.6 Conclusions

Capítulo 8.  Conclusions and future work

  • 8.1 Main contributions
  • 8.2 Future work

Referencias

Apéndice A.  The vector space Bn

Apéndice B.  Configurations for the nD-OPP´s obtained by the ´Test-Box´ algorithms

Apéndice C.  Some adjacencies´ properties of the configurations in the nD-OPP´s

Apéndice D.  Some characterizations of odd and even edges in the nD-OPP´s

Apéndice E.  Some characterizations of extreme and non-extreme vertices in the nD-OPP´s

Apéndice F.  Characterization of extreme vertices according their incident odd/even edges and relating them with other edges´ characterizations

Apéndice G.  Applying nD-EVM concepts together: an example in 4D space

Apéndice H.  The "L-Shaped" nD polytopes family

Apéndice I.  Extreme vertices countings for two color 2D-animations

Apéndice J.  Properties of function

Pérez Águila, R. 2006. Orthogonal polytopes: study and application. Tesis Doctorado. Ciencias de la Computación. Departamento de Computación, Electrónica, Física e Innovación, Escuela de Ingeniería y Ciencias, Universidad de las Américas Puebla. Noviembre. Derechos Reservados © 2006.