Polyhedra with Spherical Faces and Quasi-Fuchsian Fractals Kento - - PowerPoint PPT Presentation

▶
polyhedra with spherical faces and quasi fuchsian fractals
SMART_READER_LITE
LIVE PREVIEW

Polyhedra with Spherical Faces and Quasi-Fuchsian Fractals Kento - - PowerPoint PPT Presentation

Aug 23, 2022 321 likes •1.04k views

Topology and Computer 2017 Polyhedra with Spherical Faces and Quasi-Fuchsian Fractals Kento Nakamura Graduate School of Advanced Mathematical Science, Meiji University Sphairahedron sphaira- (= spherical) + -hedron (=

slide-1
SLIDE 1

Polyhedra with Spherical Faces and Quasi-Fuchsian Fractals

Kento Nakamura Graduate School of Advanced Mathematical Science, Meiji University Topology and Computer 2017

slide-2
SLIDE 2

Sphairahedron

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

slide-3
SLIDE 3

Quasi-sphere

One of the early examples of the 3-dimensional fractals

slide-4
SLIDE 4

Sphairahedron

slide-5
SLIDE 5

Sphairahedron

𝑇3 = 𝑆3 ∪ ∞ closed-ball: 𝑃1, 𝑃2, … , 𝑃𝑜 𝐵 = 𝑇3 − (𝑃1 ∪ 𝑃2 … ∪ 𝑃𝑜)

slide-6
SLIDE 6

Sphairahedron

One side of the simply connected two components of 𝐵

slide-7
SLIDE 7

Sphairahedron

One side of the simply connected two components of 𝐵

slide-8
SLIDE 8

Sphairahedron

One side of the simply connected two components of 𝐵

slide-9
SLIDE 9

Semi-Sphairahedron

One side of the simply connected three or more components of 𝐵

slide-10
SLIDE 10

Semi-Sphairahedron

One side of the simply connected three or more components of 𝐵

slide-11
SLIDE 11

Sphairahedron Group

𝑔

𝑗: 𝐽𝑜𝑤𝑓𝑠𝑡𝑗𝑝𝑜 𝑗𝑜 𝑃𝑗

𝐻 = < 𝑔

0, 𝑔 1, … , 𝑔 𝑜 >

slide-12
SLIDE 12

Tessellation by G

slide-13
SLIDE 13

Tessellation by G

slide-14
SLIDE 14

Tessellation by G

slide-15
SLIDE 15

Tessellation by G

slide-16
SLIDE 16

Tessellation by G

slide-17
SLIDE 17

Tessellation by G

slide-18
SLIDE 18

Tessellation by G

slide-19
SLIDE 19

Tessellation by G

slide-20
SLIDE 20

Tessellation by G

slide-21
SLIDE 21

Tessellation by G

slide-22
SLIDE 22

Tessellation by G

slide-23
SLIDE 23

Tessellation by G

slide-24
SLIDE 24

Tessellation by G

slide-25
SLIDE 25

Tessellation by G

slide-26
SLIDE 26

Tessellation by G

slide-27
SLIDE 27

The Limit Set of G

slide-28
SLIDE 28

Rationality and Ideality

Two properties to characterize sphairahedron If a sphairahedron is rational and ideal, 𝐻 is discrete.

slide-29
SLIDE 29

Rational Ideal Sphairahedron Group

Quasi-sphere (homeomorphic to a sphere) Sphairahedron

Semi-sphairahedron

slide-30
SLIDE 30

Rationality (Regularity)

All of the dihedral angles of edges is rational. (𝜌/𝑜 for the natural number 𝑜) 𝜌/3 𝜌/2, 𝜌/3, 𝜌/6

slide-31
SLIDE 31

Ideality

All of the edges are mutually tangent at its vertex

slide-32
SLIDE 32

Parameter Space

slide-33
SLIDE 33

Derivation of Parameter Space

Cube-type sphairahedron

slide-34
SLIDE 34

Graph Representation

∞ ∞ ∞

slide-35
SLIDE 35

Combination of Dihedral Angles

𝑜 = 3 3 3 3 3 3 3 3 3 3 3 3

To fulfill a ideality, the sum

  • f the dihedral angles at

each vertex should be π

slide-36
SLIDE 36

Combination of Dihedral Angles

2 3 6 3 6 3 3 3 2 2 3 6

slide-37
SLIDE 37

Combination of Dihedral Angles

2 4 4 4 4 4 2 4 2 4 2 4

slide-38
SLIDE 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 𝑨𝑐 : The height of the green sphere 𝑨𝑑 : The height of the blue sphere

slide-39
SLIDE 39

Derivation of Parameter Space

Parameter space of the cube-type sphairahedron is studied by Ahara and Araki (2003) and also Ryo Kageyama (2016).

All of the dihedral angles are 𝜌/3

𝑨𝑐𝑨𝑑 < 3/4 𝑨𝑑

2 − 𝑨𝑐𝑨𝑑 < 3/4

𝑨𝑐

2 − 𝑨𝑐𝑨𝑑 < 3/4

slide-40
SLIDE 40

Rendering Technique

slide-41
SLIDE 41

Ray Tracing

Suited for parallel computing by GPU

slide-42
SLIDE 42

Ray Tracing Eye

slide-43
SLIDE 43

Ray Tracing Eye

We have to compute an intersection between the ray and many sphairahedra

slide-44
SLIDE 44

Ray Marching Eye

Find intersection between the ray and objects

slide-45
SLIDE 45

Ray Marching Eye

slide-46
SLIDE 46

Ray Marching Eye

slide-47
SLIDE 47

Ray Marching Eye

slide-48
SLIDE 48

Ray Marching Eye

slide-49
SLIDE 49

Ray Marching Eye

slide-50
SLIDE 50

Ray Marching Hit Eye

slide-51
SLIDE 51

Sphere Tracing

slide-52
SLIDE 52

Sphere Tracing

Compute minimum distance to objects

slide-53
SLIDE 53

Sphere Tracing

slide-54
SLIDE 54

Sphere Tracing

slide-55
SLIDE 55

Sphere Tracing

slide-56
SLIDE 56

Sphere Tracing

slide-57
SLIDE 57

Sphere Tracing

slide-58
SLIDE 58

Sphere Tracing

slide-59
SLIDE 59

Sphere Tracing

slide-60
SLIDE 60

Sphere Tracing

slide-61
SLIDE 61

Sphere Tracing Hit

slide-62
SLIDE 62

Distance Function

𝐷 𝑠

A function returning the minimum distance between given point and object’s surface 𝑔 𝑞 = 𝑒𝑗𝑡𝑢𝑏𝑜𝑑𝑓 𝑞, 𝐷 − 𝑠

𝑞

slide-63
SLIDE 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; }

slide-64
SLIDE 64

Ray Tracing Eye

We need the distance to the surface

  • f the fractal
slide-65
SLIDE 65

Distance Field for the orbit of spheres

𝐷 𝑄

slide-66
SLIDE 66

Distance Field for the orbit of spheres

Inversion in 𝐷

𝐷

We need minimum distance between the point and spheres

𝑄

𝑒

slide-67
SLIDE 67

Distance Field for the orbit of spheres

𝑒

slide-68
SLIDE 68

Distance Field for the orbit of spheres

Inversion in 𝐷

𝑒

slide-69
SLIDE 69

Distance Field for the orbit of spheres

𝑒 𝑒′

slide-70
SLIDE 70

Distance Field for the orbit of spheres

𝑒 ≈ 𝑒′ 𝐾𝑏𝑑𝑝𝑐𝑗𝑏𝑜 𝑝𝑔 𝐽𝑜𝑤𝐷 𝑒′

slide-71
SLIDE 71

Experimental Sphairahedron Renderer

  • https://soma-arc.net/SphairahedronExperiment/
  • Environment … JavaScript + WebGL2.0
  • Some parameters may require high GPU Power
  • Source code

https://github.com/soma-arc/SphairahedronExperiment