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An Introduction to Computational Geometry: Polyhedra

This chapter provides an introduction to polyhedra in computational geometry, including their definition, properties, and various types such as convex polyhedra, regular polyhedra, and higher-dimensional polytopes.

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An Introduction to Computational Geometry: Polyhedra

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  1. An Introduction to Computational Geometry:Polyhedra Joseph S. B. Mitchell Stony Brook University Chapter 6: Devadoss-O’Rourke

  2. Polyhedra Definition: Solid (closed) region whose boundary is a union of a finite number of (2D) convex polygons, subject to some conditions

  3. Polyhedra vertices (0-faces), edges (1-faces), faces/facets (2-faces) • Definition: Solid (closed) region whose boundary is a union of a finite number of (2D) convex polygons, subject to some conditions:

  4. Polyhedra

  5. Polyhedra

  6. Convex Polyhedra

  7. Polyhedra: Combinatorics

  8. More General Surfaces Topological invariants of a surface S, homeomorphic to a polyhedron

  9. Proof: By induction on genus

  10. Example/Exercise

  11. Platonic Solids: Regular Convex Polyhedra in 3D Generalize the notion of a “regular polygon” (2D) Euclid, Elements (Book XIII)

  12. Platonic Solids: Regular Convex Polyhedra in 3D Faces: regular k-gons Sum of k interior angles= Thus, each interior angle= Vertex degrees = m

  13. Platonic Solids: Regular Convex Polyhedra in 3D

  14. More General “Regular” Polyhedra Allow facets that are different regular polygons, but still require vertices to “look the same”: Archimedean polyhedra (13 of them) Example: truncated icosahedron (12 pentagons, 20 hexagons): “soccer ball”

  15. More General “Regular” Polyhedra Allow facets that are different regular polygons, and allow nonconvex: uniform polyhedra (75 of them) Example: great dodecahedron

  16. 4D Polytopes Project to 3D and show the “wire diagram”: Schlegel diagram

  17. 4D Regular Polytopes • 6 regular 4D polytopes: • 4-simplex (“tetrahedron”) • hypercube (“cube”) • 4-orthoplex, or cross polytope (“octohedron”) • 24-cell • 120-cell • 600-cell

  18. d-D Regular Polytopes • 3 regular d-dimensional polytopes, d≥5: • d-simplex (“tetrahedron”) • hypercube (“cube”) • d-orthoplex, or cross polytope (“octohedron”)

  19. Convex Hull in 3D

  20. Convex Hull in 3D

  21. Data Structures Winged-edge Quad-edge DCEL

  22. Winged Edge Data Structure e1- e1+ v1 f1 e f0 v0 e0+ e0-

  23. CH in Higher Dimensions applet merge h= O(n) • 3D: Divide and conquer: • T(n)  2T(n/2) + O(n) • O(n log n) • Output-sensitive: O(n log h) [Chan] • Higher dimensions: (d  4) • O(n d/2  ), which is worst-case OPT, since point sets exist with h=(n d/2  ) • Output-sensitive: O((n+h) logd-2 h), for d=4,5 Qhull website

  24. Convex Hull in 3D

  25. Convex Hull in 3D

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