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

Polyhedra Definition: Solid (closed) region whose boundary is a union of a finite number of (2D) convex polygons, subject to some conditions: 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 4

Polyhedra 5

Convex Polyhedra 6

Polyhedra: Combinatorics 7

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

Proof: By induction on genus 9

Example/Exercise 10

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

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

Platonic Solids: Regular Convex Polyhedra in 3D 13

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

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

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

4D Regular Polytopes 6 regular 4D polytopes: 6 regular 4D polytopes: 4-simplex (“tetrahedron”)4-simplex (“tetrahedron”) hypercube (“cube”)hypercube (“cube”) 4-orthoplex, or cross polytope (“octohedron”)4-orthoplex, or cross polytope (“octohedron”) 24-cell24-cell 120-cell120-cell 600-cell600-cell 17

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

Convex Hull in 3D 19

Convex Hull in 3D 20

Data Structures Winged-edge Winged-edge Quad-edge Quad-edge DCEL DCEL 21

Winged Edge Data Structure 22 e e0-e0- e0+e0+ e1-e1- e1+e1+ v0v0 f1f1 f0f0 v1v1

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

Convex Hull in 3D 24

Convex Hull in 3D 25