Tiling Three-Dimensional Space with Simplices. Shankar Krishnan AT&T Labs - Research

Transcription

1 Tiling Three-Dimensional Space with Simplices Shankar Krishnan AT&T Labs - Research

2 What is a Tiling? Partition of an infinite space into pieces having a finite number of distinct shapes usually Euclidean space finiteness under invariant operators prototile

3 Tiling Examples Tiling with 3 prototiles Escherization tiling with single tile [Kaplan et al 00]

4 Tiling Periodic vs Aperiodic Difference arises in whether there is translational symmetry in the tiling if there is, it results in a lattice Hexagonal tiling Penrose tiling (5-fold symmetry) Aperiodic tiling [Kaplan et al 00]

5 Why study Tilings? Vector quantization Information theory source coding Wavelets, compression, blue-noise sampling Crystallography Art work Dynamical systems, non-commutative geometry Mesh generation Unstructured meshes - shapes are simplices

6 Unstructured Meshing Essential preprocessing step Simulation of physical phenomena: realistic animation in Computer Graphics, mechanics, fluids often modeled as PDE domain discretization + finite elements Important requirement: Quality Shape of simplices

7 Approximation Theory Linear interpolation: optimal shape of an element related to the Hessian of f [Shewchuk] As a general rule, best simplex - regular

8 What is a Simplex? In d-dimensions, it is the convex hull of (d+1) points in general position Simplest polytope in d-dimensions d + 1 Boundary has simplices of k + 1 dimension k 1D line segment 2D triangle 3D - tetrahedron

9 Tiling with Simplices Loosely called triangulations By definition, they are convex useful property in certain applications Algorithmic advantages periodic tilings can be generated from points Can also generate aperiodic tilings modified Penrose, Danzer tiling

10 Simple Tiling Strategy Generate points in the space Connect points to generate simplices triangulation algorithms Delaunay triangulation: canonical scheme Quality of the simplices Location of points Type of triangulation algorithm Strategy can be used to tile entire space or a portion of space

11 Triangulation Triangulation 1 Triangulation 2 How do you choose the triangulation algorithm?

12 Voronoi Diagram Given a collection of geometric primitives, it is a subdivision of space into cells such that all points in a cell are closer to one primitive than to any other Georgy Voronoi VR i { } x d x x < d x x i j = (, ) (, ), i j

13 Voronoi Diagram Voronoi Site Voronoi Region

14 Voronoi Diagram as Lower Envelopes of Distance Functions Perspective Parallel, top view Slide courtesy: Hoff et al, Siggraph 97

15 Voronoi Diagram: Demo

16 Delaunay Triangulation Dual of Voronoi diagram Graph-theoretic, topological, combinatorial Connect two Voronoi sites if they share a (d-1)-dimensional face Boris Nikolaevich Delone ( )

17 Delaunay Triangulation: Demo

18 Delaunay Triangulation Canonical, unique to any point set. Slide courtesy Alliez et al, Siggraph 05

19 Delaunay Triangulation Circumsphere of each simplex is empty Optimal in many aspects: in 2D: maximizes minimum angle Slide courtesy Alliez et al, Siggraph 05

20 Another Useful Property In 2D: lift vertex (x, y) to paraboloid (x, y, x 2 + y 2 ) Delaunay triangulation lifts to lower facets of convex hull optimal PL interpolation of isotropic paraboloid function Slide courtesy Alliez et al, Siggraph 05

21 Popular Quality Measures Radius ratio Circumradius over inradius (R/r) Aspect ratio Circumradius over shortest edge (R/e) Maximum angle (α) Dihedral angle in three dimensions important for gradient approximations most difficult to bound

22 Quality of Simplices R R α e r r α e Bounded quality measures usually implies regularity

23 Vertex Positions vs Quality Quality of the triangles is determined by the vertex positions

24 Tiling Three-Dimensional Space If allowed a single prototile, only cube and truncated octahedron tile 3-space

25 Regular Tetrahedron Unlike 2D, regular tetrahedron does not tile three-dimensional space (360 / = 5.1) dihedral angle of the regular tetrahedron = arcos(1/3) 70.53

26 Optimizing Vertex Placement Improve compactness of Voronoi cells by minimizing: 2 E = ρ( x) x x dx j= 1.. k x R j j centroid Necessary condition for optimality: Centroidal Voronoi Tessellation (CVT) site centroid

27 Optimizing Vertex Placement CVT [Du-Wang 03] best PL approximant compact Voronoi cells isotropic sampling optimized dual Best L1 approximation of paraboloid

28 In Three Dimensions Strategies like CVT do not avoid all bad simplices

29 Sliver Four well-spaced vertices near the equator of their circumsphere Bounded aspect ratio

30 Tetrahedral Close-Packed Structures (TCP) Transition metal alloys Atoms arranged as nearly regular tetrahedra [Shoemaker & Shoemaker] Frank and Kasper Studied sphere packings and dual tetrahedral networks to explain crystal structures Frank-Kasper phases characterize a catalog of TCP structures

31 Frank-Kasper Phases 14 vertices 24 tetrahedra 12 vertices 20 tetrahedra 15 vertices 26 tetrahedra 16 vertices 28 tetrahedra

32 TCP Definition Combinatorial definition [Sullivan] Triangulations whose edges have valence (number of faces adjacent to an edge) is 5 or 6, but no two edges of a triangle can have valence 6 Definition allows generation of a continuous family of TCP structures we take advantage of this to generate our near-regular tiling

33 Lattice In 3D, regular placement of points in an infinite integer grid defined by three basis vectors, V = ( v v ) 1 2 v3 any point can be expressed as av 1 + bv2 + cv3, a, b, c Lattice is a discrete subgroup of invariant to scaling and rigid transformations Represented as αrvz 3 + t Z R 3

34 Tilings with TCP Start with any tiling of the plane with equilateral triangles and square of side length 4 Color points on the boundary and interior of triangles and squares with three colors consistently. Place Red vertices at height (4i-1) Blue vertices at height (4i+1) Green vertices at height 4i and (4i+2)

35 Tilings with TCP

36 Our Tiling Scheme Start with the BCC lattice { 3} { 3 IZ U IZ + (0.5,0.5,0.5)} Nodes are given four colors: 0,1,2 or 3 p = (x,y,z) is colored i if 2( x + y + z ) i% 4 Replace nodes of color 2 and 3 by nodes of color 2.5 located at the midpoint of shortest segments connecting vertices of color 2 and 3

37 Coloring Vertices

38 Generating 2.5 Colored Vertices

39 Our Tiling Scheme Collect vertices with color 0, 1 and 2.5 Compute the Delaunay triangulation Produces tiling with almost regular tetrahedra Dual Voronoi cells all have 16 faces (12 pentagonal and 4 hexagonal) Vertex coordinates are rational

40 Our Tiling

41 Tiling Normalized Radius Ratios

42 Tiling Normalized Aspect Ratios

43 Tiling Dihedral Angles

44 Statistics of Tiling All dihedral angles between 60 and 74.2 Best range we can hope for [60-72 ] All radius ratios are > 0.97 Aspect ratios lie between [1-1.16] Best known tiling in all quality measures

45 Finite Domains

46 Finite Domains Radius ratio distribution Dihedral angle distribution

47 Acute triangulation of Square Many triangulations exist [Eppstein] What about cube? It is still open.

48 Conclusion Tiling in three dimensions a lot harder than two dimensions Existing techniques like optimization do not seem to produce adequate results TCP structure provides some insights into the problem Generate best known tiling Can we prove that it is best or do better?

49 Thank you Contact info