Visualization of Hyperbolic Tessellations

Transcription

1 Visualization of Hyperbolic Tessellations Jakob von Raumer April 9, 2013 KARLSRUHE INSTITUTE OF TECHNOLOGY KIT University of the State of Baden-Wuerttemberg and National Laboratory of the Helmholtz Association

2 Escher s Art Figure: Circle Limit III and Circle Limit IV by M. C. Escher, 1959 and 1960 Jakob von Raumer Visualization of Hyperbolic Tessellations April 9, /29

3 Escher s Art Escher s works were inspired by illustrations in a book by H.S.M. Coxeter He used woodcuts to replicate the tiles To his son George: I had an enthusiastic letter from Coxeter about my colored fish, which I sent him. Three pages of explanation of what I actually did.... It s a pity that I understand nothing, absolutely nothing of it. Jakob von Raumer Visualization of Hyperbolic Tessellations April 9, /29

4 Aims of my thesis Document and summarize theoretical basics of hyperbolic tessellations Construct suitable tiles Implement algorithms to replicate those Jakob von Raumer Visualization of Hyperbolic Tessellations April 9, /29

5 Comparison: Euclidean and hyperbolic Geometry Euclidean For a line g there is exactly one line parrallel to g that contain a point p / g. Each triangle has an angle sum of π. Hyperbolic For a line g there are more than one lines parallel to g that contain a point p / g. Each triangle has an angle sum of < π. Jakob von Raumer Visualization of Hyperbolic Tessellations April 9, /29

6 Comparison: Euclidean and hyperbolic Geometry Length of a path γ : [0, 1] H := {z C I(z) > 0}: L(γ) = Euclidean 1 0 γ (t) dt L(γ) = Hyperbolic 1 0 γ (t) I(γ(t)) dt The distance of two points a, b H is the length of the shortest path between them. Euclidean d(a, b) = b a Hyperbolic d(a, b) = ln a b + a b a b a b Jakob von Raumer Visualization of Hyperbolic Tessellations April 9, /29

7 Geodesics on the upper half-plane I g a b R 1 Figure: Geodesics on the upper half-plane. Jakob von Raumer Visualization of Hyperbolic Tessellations April 9, /29

8 From the upper half-plane to the Poincaré disk model The contiuous map f : H U : z zi+1 z+i induces a bounded presentation of H. I I f R R 1 Figure: Geodesics on the upper half-plane and their correspondents on the Poincaré disk model. Jakob von Raumer Visualization of Hyperbolic Tessellations April 9, /29

9 The Beltrami-Klein model g Figure: A polygon, shown in the Poincaré disk model and in the Beltrami-Klein model Jakob von Raumer Visualization of Hyperbolic Tessellations April 9, /29

10 Isometries on H Theorem The isometries on H are a group isomorphic to PS L(2, R) := S L(2, R)/ {±I 2 }. The orientation preserving isometries on H are isomorphic to PSL(2, R) := SL(2, R)/ {±I 2 }. ( a b c d ) PSL(2, R) corresponds to the Möbius transformation z az+b cz+d Examples: Translation z z + 1 Dilation z 2z Rotation z 1 z Jakob von Raumer Visualization of Hyperbolic Tessellations April 9, /29

11 3 types of transformations ( a b ) c d is trace fixed points figure elliptic a + d < 2 one in H parabolic a + d = 2 hyperbolic a + d > 2 one at infinity two at infinity Jakob von Raumer Visualization of Hyperbolic Tessellations April 9, /29

12 3 types of transformations ( a b c d ) is trace fixed points figure elliptic a + d < 2 one in H parabolic a + d = 2 one at infinity hyperbolic a + d > 2 two at infinity Jakob von Raumer Visualization of Hyperbolic Tessellations April 9, /29

13 3 types of transformations ( a b c d ) is trace fixed points figure elliptic a + d < 2 one in H parabolic a + d = 2 one at infinity hyperbolic a + d > 2 two at infinity Jakob von Raumer Visualization of Hyperbolic Tessellations April 9, /29

14 Fuchsian and Kleinian groups A discrete subgroup Γ of PS L(2, R), is called Kleinian group. If additionally Γ PSL(2, R), then it s called Γ Fuchsian group. Jakob von Raumer Visualization of Hyperbolic Tessellations April 9, /29

15 Fundamental domains Definition A closed subset F H is a fundamental domain for Γ, iff: Γ F := T Γ T (F) = H. For all T Γ, F and T (F) intersect only in their boundary. If F is a fundamental domain for Γ, then {T (F) T Γ} is called a tessellation. Jakob von Raumer Visualization of Hyperbolic Tessellations April 9, /29

16 Fuchsian Groups: Elliptic and parabolic subgroups Elliptic subgroup: T Γ where T is elliptic Parabolic subgroup: T Γ where T is parabolic Maximal elliptic or parabolic subgroups which are conjugate to each other, have the same order. They are called periods of Γ Jakob von Raumer Visualization of Hyperbolic Tessellations April 9, /29

17 The orbit space of a Fuchsian group Jakob von Raumer Visualization of Hyperbolic Tessellations April 9, /29

18 The signature of a Fuchsian group Definition Let Γ be a Fuchsian group with periods m 1,..., m n N 0 { }, m 1... m n and genus g N 0. Then the vector (g, m 1,..., m n ) is called the signature of Γ. Jakob von Raumer Visualization of Hyperbolic Tessellations April 9, /29

19 The program s features The program should create tessellations which are induced by a Fuchsian group with a given signature or consists of polygons with a given sequence of inner angles 2π m i. Jakob von Raumer Visualization of Hyperbolic Tessellations April 9, /29

20 Polygons for tiling by reflections β i = 2π m i lim r 0 θ i = π lim r 1 θ i = 0 Find r 0 (0, 1) such that n i+0 θ i = 2π. I 0.2 c θi b b r a βi R d 0.4 Jakob von Raumer Visualization of Hyperbolic Tessellations April 9, /29

21 Polygons for tiling by Fuchsian groups β 1 v9 α2 v8 v10 α 1 β2 v11 v7 t β1 α 2 v6 v12 α1 β 2 v5 s4 s3 s1 v1 ξ 4 w4 s2 ξ1 ξ4 v4 v3 v2 ξ 1 w1 ξ 3 w3 ξ3 ξ 2 ξ2 w2 Motivation Figure: Construction Hyperbolicof Geometry the polygon for g = 2 and Drawing n = hyperbolic 4 tessellations Jakob von Raumer Visualization of Hyperbolic Tessellations April 9, /29

22 Replication algorithm by Dunham It s based on a depth-first search. It s a combinatorial algorithm: Approach only depends on the corner valencies of the polygon. It can replicate arbitrary polygons, except for: Triangular fundamental domains or at least one corner valency of three. Jakob von Raumer Visualization of Hyperbolic Tessellations April 9, /29

23 Search trees of the Dunham algorithm Jakob von Raumer Visualization of Hyperbolic Tessellations April 9, /29

24 Separating the tessellation into layers Jakob von Raumer Visualization of Hyperbolic Tessellations April 9, /29

25 Search trees of the Dunham algorithm Jakob von Raumer Visualization of Hyperbolic Tessellations April 9, /29

26 Repliction using a priority queue Basic approach: Transform each tile by all transformations mapping it to edge adjacent tiles Discard tiles we already met Jakob von Raumer Visualization of Hyperbolic Tessellations April 9, /29

27 Repliction using a priority queue Three data structures: Liste inactivepolys: Polygons yet expanded Priority queue activepolys: Polygons still to be expanded Hash set midpoints: Midpoints of polygons we met so far Jakob von Raumer Visualization of Hyperbolic Tessellations April 9, /29

28 Repliction using a priority queue Three data structures: Liste inactivepolys: Polygons yet expanded Priority queue activepolys: Polygons still to be expanded Hash set midpoints: Midpoints of polygons we met so far Jakob von Raumer Visualization of Hyperbolic Tessellations April 9, /29

29 Repliction using a priority queue Three data structures: Liste inactivepolys: Polygons yet expanded Priority queue activepolys: Polygons still to be expanded Hash set midpoints: Midpoints of polygons we met so far Jakob von Raumer Visualization of Hyperbolic Tessellations April 9, /29

30 Comparing the run time Run time in milliseconds Dunham Algorithm Priority Queue Algorithm x 3 50 x Number of polygons Jakob von Raumer Visualization of Hyperbolic Tessellations April 9, /29

31 Comparing the run time 10 4 Dunham Algorithm Priority Queue Algorithm Run time in milliseconds Number of edges Jakob von Raumer Visualization of Hyperbolic Tessellations April 9, /29

32 Outlook Questions and Work still to accomplish: Try to solve the restrictions of the Dunham algorithm Optimizing the approximation used in the creation of the base polygons Replicate arbitrary vector graphics on the base polygons Interactive zoom and pan Adapting the stroke width according to hyperbolic geometry Jakob von Raumer Visualization of Hyperbolic Tessellations April 9, /29

33 On something completely unrelated... obw.ouinternational.ca Jakob von Raumer Visualization of Hyperbolic Tessellations April 9, /29