Polyhedra with Spherical Faces and Quasi-Fuchsian Fractals

Transcription

1 Topology and Computer 2017 Polyhedra with Spherical Faces and Quasi-Fuchsian Fractals Kento Nakamura Graduate School of Advanced Mathematical Science, Meiji University

2 Sphairahedron sphaira- (= spherical) + -hedron (= polyhedron) New geometrical concept invented by Kazushi Ahara and Yoshiaki Araki (2003)

3 Quasi-sphere One of the early examples of the 3-dimensional fractals

4 Sphairahedron

5 Sphairahedron S 3 = R 3 closed-ball: O 1, O 2,, O n A = S 3 (O 1 O 2 O n )

6 Sphairahedron One side of the simply connected two components of A

7 Sphairahedron One side of the simply connected two components of A

8 Sphairahedron One side of the simply connected two components of A

9 Semi-Sphairahedron One side of the simply connected three or more components of A

10 Semi-Sphairahedron One side of the simply connected three or more components of A

11 Sphairahedron Group f i : Inversion in O i G = < f 0, f 1,, f n >

12 Tessellation by G

13 Tessellation by G

14 Tessellation by G

15 Tessellation by G

16 Tessellation by G

17 Tessellation by G

18 Tessellation by G

19 Tessellation by G

20 Tessellation by G

21 Tessellation by G

22 Tessellation by G

23 Tessellation by G

24 Tessellation by G

25 Tessellation by G

26 Tessellation by G

27 The Limit Set of G

28 Rationality and Ideality Two properties to characterize sphairahedron If a sphairahedron is rational and ideal, G is discrete.

29 Rational Ideal Sphairahedron Group Sphairahedron Semi-sphairahedron Quasi-sphere (homeomorphic to a sphere)

30 Rationality (Regularity) All of the dihedral angles of edges is rational. (π/n for the natural number n) π/3 π/2, π/3, π/6

31 Ideality All of the edges are mutually tangent at its vertex

32 Parameter Space

33 Derivation of Parameter Space Cube-type sphairahedron

34 Graph Representation

35 Combination of Dihedral Angles n = To fulfill a ideality, the sum of the dihedral angles at each vertex should be π

36 Combination of Dihedral Angles

37 Combination of Dihedral Angles

38 Derivation of Parameter Space Fix prism and a sphere The prism is inscribed inside an unit circle. The height of the red sphere is 0. Parameter z b : The height of the green sphere z c : The height of the blue sphere

39 Derivation of Parameter Space All of the dihedral angles are π/3 z b z c < 3/4 z c 2 z b z c < 3/4 z b 2 z b z c < 3/4 Parameter space of the cube-type sphairahedron is studied by Ahara and Araki (2003) and also Ryo Kageyama (2016).

40 Rendering Technique

41 Ray Tracing Suited for parallel computing by GPU

42 Ray Tracing Eye

43 Ray Tracing We have to compute an intersection between the ray and many sphairahedra Eye

44 Ray Marching Find intersection between the ray and objects Eye

45 Ray Marching Eye

46 Ray Marching Eye

47 Ray Marching Eye

48 Ray Marching Eye

49 Ray Marching Eye

50 Ray Marching Eye Hit

51 Sphere Tracing

52 Sphere Tracing Compute minimum distance to objects

53 Sphere Tracing

54 Sphere Tracing

55 Sphere Tracing

56 Sphere Tracing

57 Sphere Tracing

58 Sphere Tracing

59 Sphere Tracing

60 Sphere Tracing

61 Sphere Tracing Hit

62 Distance Function A function returning the minimum distance between given point and object s surface f p = distance p, C r p r C

63 Distance to Sphairahedron float DistanceToSphairahedron (vec3 p) { float d = DistanceToPrism(p); d = max(-distancetospherea(p), d); d = max(-distancetosphereb(p), d); d = max(-distancetospherec(p), d); return d; }

64 Ray Tracing We need the distance to the surface of the fractal Eye

65 Distance Field for the orbit of spheres C P

66 Distance Field for the orbit of spheres C Inversion in C P d We need minimum distance between the point and spheres

67 Distance Field for the orbit of spheres d

68 Distance Field for the orbit of spheres d Inversion in C

69 Distance Field for the orbit of spheres d d

70 Distance Field for the orbit of spheres d d d Jacobian of InvC

71 Experimental Sphairahedron Renderer Environment JavaScript + WebGL2.0 Some parameters may require high GPU Power Source code