An Introduction to Computational Geometry:

Polyhedra

Joseph S. B. Mitchell Stony Brook University

Chapter 6: Devadoss-O’Rourke

Polyhedra

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

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Polyhedra

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

•

• 3

vertices (0-faces), edges (1-faces), faces/facets (2-faces)

Polyhedra

•

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Polyhedra

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Convex Polyhedra

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Polyhedra: Combinatorics

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More General Surfaces

Topological invariants of a surface S, homeomorphic to a polyhedron

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Proof: By induction on genus

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Example/Exercise

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Platonic Solids: Regular Convex Polyhedra in 3D

Generalize the notion of a “regular polygon” (2D)

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Euclid, Elements (Book XIII)

Platonic Solids: Regular Convex Polyhedra in 3D

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Faces: regular k-gons Sum of k interior angles= Thus, each interior angle= Vertex degrees = m

Platonic Solids: Regular Convex Polyhedra in 3D

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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”

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More General “Regular” Polyhedra

Allow facets that are different regular polygons, and allow nonconvex: uniform polyhedra (75 of them)

Example: great dodecahedron

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4D Polytopes

Project to 3D and show the “wire diagram”: Schlegel diagram

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4D Regular Polytopes

6 regular 4D polytopes: • 4-simplex (“tetrahedron”)

• hypercube (“cube”)

• 4-orthoplex, or cross polytope (“octohedron”)

• 24-cell

• 120-cell

• 600-cell

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d-D Regular Polytopes

3 regular d-dimensional polytopes, d≥5: • d-simplex (“tetrahedron”)

• hypercube (“cube”)

• d-orthoplex, or cross polytope (“octohedron”)

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Convex Hull in 3D

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Convex Hull in 3D

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Data Structures

Winged-edge

Quad-edge

DCEL

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Winged Edge Data Structure

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e

e0-

e0+

e1- e1+

v0

f1

f0

v1

CH in Higher Dimensions

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

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merge

h= O(n)

Qhull website

applet

http://www.qhull.org/http://www.cse.unsw.edu.au/~lambert/java/3d/hull.html?dimension=2D&model=Gift+Wrap&seed=45123&npoints=20&frame=1&labels=15

Convex Hull in 3D

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Convex Hull in 3D

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