Optimizing triangular meshes to have the incrircle packing property

Transcription

1 Packing Circles and Spheres on Surfaces Ali Mahdavi-Amiri

2 Introduction Optimizing triangular meshes to have p g g the incrircle packing property

3 Our Motivation PYXIS project Geometry Nature

4 Geometry Isoperimetry Uniform connectivity Additional equidistant t neighbors Densest Circle Packing

5 Nature Equal Area Honeycomb Retina Middleton, Lee,Sibaswamy, Jayanthi. Hexagonal Image Processing, A practical Approach. Springer Press.

6 Applications Architectural Projects Different Surfaces

7 Other Motivations interesting gproblem in pure geometry; Hilbert s list connections with lattice theory and group theory applications to number theory digital communications: error- correcting codes

8 Densest Packing circles in 2D δ = π 12 Hexagonal Lattice

9 Packing Circles in 3D δ = π 12 Gauss; the face-centered cubic packing Honey comb

10 Hilbert s 18 problem

11 CP Meshes Orange incircles form a packing g p g Blue spheres form a packing

12 Conformal Geometry and mapping Conformal geometry: set of angle-preserving transformations A conformal map : function which preserves the angle.

13 Mobius transformation A Mobius transformation of the plane is a rational function of the form: Mobius transformation is bijective and conformal.

14 Mobius Geometry Mobius Geometry is the study of Euclidean space with a point added at infinity. M Mobius transformation

15 Hybrid Meshes A hybrid mesh is a mesh that contains structured portions and unstructured portions.

16 Combinatoric Stephenson,k Introduction to Circle Packing. Cambridge University. Press.

17 Main Results optimization of CP meshes & conformal geometry Hybrid meshes suitable for architecture Approximate circle pickings and patterns on arbitrary freeform shapes

18 Triangle meshes with an incircle packing

19 Associated Packing of Spheres The distance between a vertex and all its contact points must be equal. r : radius of the sphere. Sphere packing is orthogonal to the circle packing.

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21 Packing as a Mobius object CP mesh by circles and spheres Mobius maps: circle to circle sphere to sphere Mobius maps: CP meshes to CP meshes.

22 Packing circles No guarantee to have a packing circle g p g for all meshes.

23 Irregular CPs We can have PCs with different rs for different vertices.

24 Two triangles with one edge Two triangles and : We have: Two incircles meet??

25 Two incircles meet?

26 Conic C

27 Circles and spheres and are co planar. are coplanar. Projecting one circle from c. c

28 Quads and incircles

29 Optimization Algorithm incricle packing property proximity of the mesh to the reference surface proximity of the mesh boundary to boundary curve

30 Optimization Algorithm

31 Optimization Algorithm

32 Optimization Algorithm : nonlinear least square problem for vertex locations We have as two representation and. Damped Gauss-newton Cholmoid for Cholesky factorization.

33 Optimization Algorithm : topological disk or sphere Arbitrary triangle mesh

34 Arbitrary Triangle Meshes

35 Arbitrary Triangle Meshes

36 Solvability of optimization problem Start not necessarily CP Not cp Not cp Conformal Conformal mapping mapping cp Optimization succeeds

37 Genus Number of Handles

38 Solvability of optimization problem : closed surface of genus g. Genus 0 (topological sphere) Genus>0 unless initial mesh is adapted to

39 Initial Mesh Using Centroidal Voronoi Diagram. Computed by Lioyd relaxation Isotropic Mesh Random

40 Voronoi Diagram Decomposition of a space based on a set of points Centroidal: generating point of each cell is its mean.

41 Lioyd Relaxation Initial distribution of points 1. Voronoi diagram of all points 2. Computing the centroid of each cell 3. Each point moved to each centroid yes 4. If lattice is isotropic i End no Another set of points go to 1.

42 Solvability of optimization problem is homeomorphic to a planar domain with b boundaries topological disk b=1 Annulus b=2 r b>2 Adapted to surface

43 Solvability of optimization problem g, b>0 Riemann Surface Riemann surface X For every x of X there is a neighborhood containing x homeomorphic to a disk in complex plane.

44 Solvability of optimization problem

45 Derived structures Tri hex octagon

46 Derived structures

47 Torsion free beam layouts

48 Filling Hexagons Quads (inserting one point)

49 Circle packing and patterns

50 Packing of circles and arc Splines

51 Almost-rhombic A mesh where each quad obeys:

52 Derived structures Original mesh Interior view Close up Close up

53 Results CP meshes variety of shapes Hybrid meshes are better to cover different surfaces in comparison with pure quads or hex. We can substitute different shapes easily. Appealing for interior design

54 Results Free form circles Louis Vuitton Store Paris

55 Limitations & Future work Success of optimization Defining Mobius geometry on CP meshes (defining the ratio) Relation to conformal Geometry Higher genus and boundaries

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