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conformal equivalence of triangle meshes

Conformal Equivalence of Triangle Meshes Boris Springborn with - - PowerPoint PPT Presentation

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Conformal Equivalence of Triangle Meshes Boris Springborn with Peter Schrder und Ulrich Pinkall DFG Research Center M ATHEON Mathematics for key technologies Obergurgl, 17. Dezember 2008 introduction 1 smooth discrete 2 some basics 3

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SLIDE 1

Conformal Equivalence

f Triangle Meshes

Boris Springborn with Peter Schröder und Ulrich Pinkall

DFG Research Center MATHEON Mathematics for key technologies

Obergurgl, 17. Dezember 2008

slide-2

SLIDE 2

1

introduction

2

smooth → discrete

3

some basics

4

discrete mapping problem

5

variational principle

6

piecewise projective interpolation

7

hyperbolic geometry

Conformal Equivalence of Triangle Meshes 2 / 32

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SLIDE 3

Introduction

1 discrete version of conformal maps 2 polyhedral realization of hyperbolic

cusp metrics

Conformal Equivalence of Triangle Meshes 3 / 32

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SLIDE 4

Introduction

1 discrete version of conformal maps 2 polyhedral realization of hyperbolic

cusp metrics

Conformal Equivalence of Triangle Meshes 3 / 32

slide-5

SLIDE 5

Conformal maps

conformal means angle preserving infinitesimal lengths scaled by conformal factor |df | = eu |dx| independent of direction in the small like similarity transformations Problem: surface in space

conformally

− − − − − − → plane

Conformal Equivalence of Triangle Meshes 4 / 32

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SLIDE 6

Example: Mercator’s projection (1569)

Conformal Equivalence of Triangle Meshes 5 / 32

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SLIDE 7

Example: Mercator’s projection (2008)

Conformal Equivalence of Triangle Meshes 6 / 32

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SLIDE 8

Example: Mercator’s projection (2008)

Conformal Equivalence of Triangle Meshes 6 / 32

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SLIDE 9

Example: Mercator’s projection (2008)

Conformal Equivalence of Triangle Meshes 6 / 32

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SLIDE 10

Example: Mercator’s projection (2008)

Conformal Equivalence of Triangle Meshes 6 / 32

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SLIDE 11

Example: Mercator’s projection (2008)

Conformal Equivalence of Triangle Meshes 6 / 32

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SLIDE 12

Example: Mercator’s projection (2008)

Conformal Equivalence of Triangle Meshes 6 / 32

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SLIDE 13

Example: Mercator’s projection (2008)

Conformal Equivalence of Triangle Meshes 6 / 32

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SLIDE 14

Example: Mercator’s projection (2008)

Conformal Equivalence of Triangle Meshes 6 / 32

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SLIDE 15

Example: Mercator’s projection (2008)

Conformal Equivalence of Triangle Meshes 6 / 32

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SLIDE 16

Example: Mercator’s projection (2008)

Conformal Equivalence of Triangle Meshes 6 / 32

slide-17

SLIDE 17

Example: Mercator’s projection (2008)

Conformal Equivalence of Triangle Meshes 6 / 32

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SLIDE 18

Example: Mercator’s projection (2008)

Conformal Equivalence of Triangle Meshes 6 / 32

slide-19

SLIDE 19

Example: Mercator’s projection (2008)

Conformal Equivalence of Triangle Meshes 6 / 32

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SLIDE 20

Example: Mercator’s projection (2008)

Conformal Equivalence of Triangle Meshes 6 / 32

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SLIDE 21

Practical applications

texture mapping, remeshing want to map arbitrary surfaces, given as triangle meshes cone-like singularities can lower area distortion

Conformal Equivalence of Triangle Meshes 7 / 32

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SLIDE 22

Practical applications

texture mapping, remeshing want to map arbitrary surfaces, given as triangle meshes cone-like singularities can lower area distortion

Conformal Equivalence of Triangle Meshes 7 / 32

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SLIDE 23

Practical applications

texture mapping, remeshing want to map arbitrary surfaces, given as triangle meshes cone-like singularities can lower area distortion

Conformal Equivalence of Triangle Meshes 7 / 32

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SLIDE 24

Practical applications

texture mapping, remeshing want to map arbitrary surfaces, given as triangle meshes cone-like singularities can lower area distortion

Conformal Equivalence of Triangle Meshes 7 / 32

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SLIDE 25

Examples

Conformal Equivalence of Triangle Meshes 8 / 32

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SLIDE 26

1

introduction

2

smooth → discrete

3

some basics

4

discrete mapping problem

5

variational principle

6

piecewise projective interpolation

7

hyperbolic geometry

Conformal Equivalence of Triangle Meshes 9 / 32

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SLIDE 27

Smooth theory

Definition Two Riemannian metrics g, ˜ g on a smooth manifold M are called conformally equivalent, if ˜ g = e2u g for some function u : M → R Gaussian curvatures e2u ˜ K = K + ∆gu mapping problem ⇔ Given surface (M, g), find conformally equivalent flat metric ˜ g Poisson problem ∆gu = −K

Conformal Equivalence of Triangle Meshes 10 / 32

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SLIDE 28

Discrete

abstract surface triangulation M = (V , E, T) Definition A discrete metric on M is a function ℓ : E → R>0, ij → ℓij satifying all triangle inequalities: ∀ ijk ∈ T : ℓij < ℓjk + ℓki ℓjk < ℓki + ℓij ℓki < ℓij + ℓjk

Conformal Equivalence of Triangle Meshes 11 / 32

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SLIDE 29

Discrete

Definition Two discrete metrics ℓ, ˜ ℓ on M are (discretely) conformally equivalent if ˜ ℓij = e

1 2 (ui+uj)ℓij

for some function u : V → R use λij = 2 log ℓij so ℓij = eλij/2 and ˜ λij = λij + ui + uj

Conformal Equivalence of Triangle Meshes 11 / 32

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SLIDE 30

1

introduction

2

smooth → discrete

3

some basics

4

discrete mapping problem

5

variational principle

6

piecewise projective interpolation

7

hyperbolic geometry

Conformal Equivalence of Triangle Meshes 12 / 32

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SLIDE 31

Two single triangles

two single triangles are always conformally equivalent ˜ λ12 = λ12 + u1 + u2 ˜ λ23 = λ23 + u2 + u3 ˜ λ31 = λ31 + u3 + u1 eλ23/2 eλ31/2 eλ12/2 e˜

λ23/2

e˜

λ31/2

u2 u3 u1 e˜

λ12/2

Conformal Equivalence of Triangle Meshes 13 / 32

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SLIDE 32

Two single triangles

two single triangles are always conformally equivalent ˜ λ12 = λ12 + u1 + u2 + ˜ λ23 = λ23 + u2 + u3 + ˜ λ31 = λ31 + u3 + u1 − eλ23/2 eλ31/2 eλ12/2 e˜

λ23/2

e˜

λ31/2

u2 u3 u1 e˜

λ12/2

Conformal Equivalence of Triangle Meshes 13 / 32

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SLIDE 33

Length cross ratio

Definition For interior edges ij define length cross ratio lcrij = ℓih ℓjk ℓhj ℓki ℓ, ˜ ℓ discretely conformally equivalent

lcrij = lcrij

Conformal Equivalence of Triangle Meshes 14 / 32

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SLIDE 34

Teichmüller space

∀ interior vertices i:

ij∋i

lcrij = 1 discrete conformal strukture on M: equivalence class of discrete metrics M closed, compact, genus g: dim{conformal structures} = |E| − |V | = 6g − 6 + 2|V | = dim Tg,|V | Tg,n: Teichmüller space for genus g with n punctures

Conformal Equivalence of Triangle Meshes 15 / 32

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SLIDE 35

Möbius invariance

immersion V → Rn, i → vi induces discrete metric ℓij = vi − vj Möbius transformation: composition of inversions on spheres the only conformal transformations in Rn if n ≥ 3 Möbius equivalent immersions induce conformally equivalent discrete metrics follows from

p

p2 − q q2

=

1 p · 1 q

p − q
Conformal Equivalence of Triangle Meshes

16 / 32

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SLIDE 36

1

introduction

2

smooth → discrete

3

some basics

4

discrete mapping problem

5

variational principle

6

piecewise projective interpolation

7

hyperbolic geometry

Conformal Equivalence of Triangle Meshes 17 / 32

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SLIDE 37

Angles and curvatures

lengths determine angles αi

jk = 2 tan−1 (−ℓij+ℓjk+ℓki)(ℓij+ℓjk−ℓki) (ℓij−ℓjk+ℓki)(ℓij+ℓjk+ℓki)

angles sum around vertex i Θi =

ijk∋i

αi

jk

curvature at interior vertex i Ki = 2π − Θi boundary curvature at boundary vertex κi = π − Θi j i αi

jk

k ℓij ℓki ℓjk

Conformal Equivalence of Triangle Meshes 18 / 32

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SLIDE 38

Mapping problem

Discrete mapping problem Given mesh M, metric ℓij = e

1 2λij, and

desired angle sums Θi Find conformally equivalent metric ˜ ℓij with

Θi =

Θi

Θi = 2π for interior vertices

(except for cone-like singulatrities) if ˜ ℓij are found, lay out triangles non-linear equations for ui

Conformal Equivalence of Triangle Meshes 19 / 32

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SLIDE 39

Mapping problem

Discrete mapping problem Given mesh M, metric ℓij = e

1 2λij, and

desired angle sums Θi Find conformally equivalent metric ˜ ℓij with

Θi =

Θi

Θi = 2π for interior vertices

(except for cone-like singulatrities) if ˜ ℓij are found, lay out triangles non-linear equations for ui looks bad

Conformal Equivalence of Triangle Meshes 19 / 32

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SLIDE 40

Mapping problem

Discrete mapping problem Given mesh M, metric ℓij = e

1 2λij, and

desired angle sums Θi Find conformally equivalent metric ˜ ℓij with

Θi =

Θi

Θi = 2π for interior vertices

(except for cone-like singulatrities) if ˜ ℓij are found, lay out triangles non-linear equations for ui looks bad variational principle comes to the rescue

Conformal Equivalence of Triangle Meshes 19 / 32

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SLIDE 41

1

introduction

2

smooth → discrete

3

some basics

4

discrete mapping problem

5

variational principle

6

piecewise projective interpolation

7

hyperbolic geometry

Conformal Equivalence of Triangle Meshes 20 / 32

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SLIDE 42

Variational principle

S(u)

def

=

ijk∈T
˜

αk

ij˜

λij + ˜ αi

jk˜

λjk + ˜ αj

ki˜

λki − π(ui + uj + uk) +2 L(˜ αk

ij) + 2 L(˜

αi

jk) + 2 L(˜

αj

ki)

+
i∈V
Θiui

Milnor’s Lobachevsky function L(α) = − α log |2 sin t| dt ∂S ∂ui = Θi − Θi ˜ ℓij = e

1 2(λij+ui+uj) solves mapping problem

u = (u1, . . . , un) is critical point of S(u)

Conformal Equivalence of Triangle Meshes 21 / 32

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SLIDE 43

How does it work?

f (x1, x2, x3) = α1 x1 + α2 x3 + α3 x3 + L(α1) + L(α2) + L(α3) 1 3 2 α3 α1 α2 a2 = ex2 a3 = ex3 a1 = ex1

Conformal Equivalence of Triangle Meshes 22 / 32

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SLIDE 44

How does it work?

f (x1, x2, x3) = α1 x1 + α2 x3 + α3 x3 + L(α1) + L(α2) + L(α3) 1 3 2 α3 α1 α2 a2 = ex2 a3 = ex3 a1 = ex1 ∂f ∂x1 = α1

Conformal Equivalence of Triangle Meshes 22 / 32

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SLIDE 45

How does it work?

f (x1, x2, x3) = α1 x1 + α2 x3 + α3 x3 + L(α1) + L(α2) + L(α3) L′(α) = − log |2 sin α| 1 3 2 α3 α1 α2 a2 = ex2 a3 = ex3 a1 = ex1 ∂f ∂x1 = α1

Conformal Equivalence of Triangle Meshes 22 / 32

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SLIDE 46

How does it work?

f (x1, x2, x3) = α1 x1 + α2 x3 + α3 x3 + L(α1) + L(α2) + L(α3) L′(α) = − log |2 sin α| ∂f ∂x1 = α1 +

x1 − log(2 sin α1)

∂α1 ∂x1 +

x2 − log(2 sin α2)

∂α2 ∂x1 +

x3 − log(2 sin α3)

∂α3 ∂x1 1 3 2 α3 α1 α2 a2 = ex2 a3 = ex3 a1 = ex1 ∂f ∂x1 = α1

Conformal Equivalence of Triangle Meshes 22 / 32

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SLIDE 47

How does it work?

f (x1, x2, x3) = α1 x1 + α2 x3 + α3 x3 + L(α1) + L(α2) + L(α3) L′(α) = − log |2 sin α| ∂f ∂x1 = α1 +

x1 − log(2 sin α1)

∂α1 ∂x1 +

x2 − log(2 sin α2)

∂α2 ∂x1 +

x3 − log(2 sin α3)

∂α3 ∂x1 1 3 2 α3 α1 α2 a2 = ex2 a3 = ex3 a1 = ex1 xi = log ai = ⇒

xi − log(2 sin αi)
= log

ai 2 sin αi ∂f ∂x1 = α1

Conformal Equivalence of Triangle Meshes 22 / 32

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SLIDE 48

How does it work?

f (x1, x2, x3) = α1 x1 + α2 x3 + α3 x3 + L(α1) + L(α2) + L(α3) L′(α) = − log |2 sin α| ∂f ∂x1 = α1 +

x1 − log(2 sin α1)

∂α1 ∂x1 +

x2 − log(2 sin α2)

∂α2 ∂x1 +

x3 − log(2 sin α3)

∂α3 ∂x1 1 3 2 α3 α1 α2 a2 = ex2 a1 = ex1 a3 = ex3 R xi = log ai = ⇒

xi − log(2 sin αi)
= log

ai 2 sin αi = log R ∂f ∂x1 = α1

Conformal Equivalence of Triangle Meshes 22 / 32

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SLIDE 49

How does it work?

f (x1, x2, x3) = α1 x1 + α2 x3 + α3 x3 + L(α1) + L(α2) + L(α3) L′(α) = − log |2 sin α| ∂f ∂x1 = α1 +

x1 − log(2 sin α1)

∂α1 ∂x1 +

x2 − log(2 sin α2)

∂α2 ∂x1 +

x3 − log(2 sin α3)

∂α3 ∂x1 1 3 2 α3 α1 α2 a2 = ex2 a1 = ex1 a3 = ex3 R xi = log ai = ⇒

xi − log(2 sin αi)
= log

ai 2 sin αi = log R ∂f ∂x1 = α1 + log R ·

∂ ∂x1 (α1 + α2 + α3)

Conformal Equivalence of Triangle Meshes 22 / 32

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SLIDE 50

How does it work?

f (x1, x2, x3) = α1 x1 + α2 x3 + α3 x3 + L(α1) + L(α2) + L(α3) L′(α) = − log |2 sin α| ∂f ∂x1 = α1 +

x1 − log(2 sin α1)

∂α1 ∂x1 +

x2 − log(2 sin α2)

∂α2 ∂x1 +

x3 − log(2 sin α3)

∂α3 ∂x1 1 3 2 α3 α1 α2 a2 = ex2 a1 = ex1 a3 = ex3 R xi = log ai = ⇒

xi − log(2 sin αi)
= log

ai 2 sin αi = log R ∂f ∂x1 = α1 + log R ·

✘✘✘✘✘✘✘✘✘ ✘ ✿ 0

∂ ∂x1 (α1 + α2 + α3)

Conformal Equivalence of Triangle Meshes 22 / 32

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SLIDE 51

Convexity

S(u) =

ijk∈T
2f (

˜ λij 2 , ˜ λjk 2 , ˜ λki 2 ) − π(ui + uj + uk)

+
i∈V
Θiui

Conformal Equivalence of Triangle Meshes 23 / 32

slide-52

SLIDE 52

Convexity

S(u) =

ijk∈T
2f (

˜ λij 2 , ˜ λjk 2 , ˜ λki 2 ) − π(ui + uj + uk)

+
i∈V
Θiui

good news: S(u) strictly convex bad news: domain not convex (due to triangle inequalities)

u2 u1 u1 + u2 + u3 = 0

Conformal Equivalence of Triangle Meshes 23 / 32

slide-53

SLIDE 53

Convexity

S(u) =

ijk∈T
2f (

˜ λij 2 , ˜ λjk 2 , ˜ λki 2 ) − π(ui + uj + uk)

+
i∈V
Θiui

good news: S(u) strictly convex bad news: domain not convex (due to triangle inequalities) good news: can be extended set angles in “broken” triangles to (0, 0, π) → convex C 1 function on R|V | solution is unique (if it exists)

ne finds it by minimizing S(u)

u2 u1 u1 + u2 + u3 = 0

Conformal Equivalence of Triangle Meshes 23 / 32

slide-54

SLIDE 54

Side remark on amoebas

Let p(z1, . . . , zk) ∈ C[z1, . . . , zk] amoeba of p is Ap =

x ∈ Rk
p(z) = 0 has solution

with |z1| = ex1, . . . , |zk| = exk

[Gelfand, Kapranov, Zelevinsky]

Ronkin function for p is Rp(x) = 1 2πi k

S1(ex1)×...×S1(exk )

log |p(z)| dz1 z1 ∧ . . . ∧ dzk zk domain of f (x1, x2, x3) is Az1+z2+z3 Rz1+z2+z3(x1, x2, x3) = 1 π f (x1, x2, x3) (with linear extension)

Conformal Equivalence of Triangle Meshes 24 / 32

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SLIDE 55

Boundary conditions

Neumann fix angle sums at boundary Dirichlet fix ui at boundary u = 0 → isometry on boundary least distortion (measured by

grad u2)

Conformal Equivalence of Triangle Meshes 25 / 32

slide-56

SLIDE 56

1

introduction

2

smooth → discrete

3

some basics

4

discrete mapping problem

5

variational principle

6

piecewise projective interpolation

7

hyperbolic geometry

Conformal Equivalence of Triangle Meshes 26 / 32

slide-57

SLIDE 57

Piecewise projective interpolation

want to interpolate over triangles piecewise linear interpolation always works better: circumcircle preserving piecewise projective interpolation continuous across edges

meshes discretely conformally equivalent

looks better for coarse meshes (important for computer games) linear

Conformal Equivalence of Triangle Meshes 27 / 32

slide-58

SLIDE 58

Piecewise projective interpolation

want to interpolate over triangles piecewise linear interpolation always works better: circumcircle preserving piecewise projective interpolation continuous across edges

meshes discretely conformally equivalent

looks better for coarse meshes (important for computer games)

Conformal Equivalence of Triangle Meshes 27 / 32

slide-59

SLIDE 59

Piecewise projective interpolation

want to interpolate over triangles piecewise linear interpolation always works better: circumcircle preserving piecewise projective interpolation continuous across edges

meshes discretely conformally equivalent

looks better for coarse meshes (important for computer games)

Conformal Equivalence of Triangle Meshes 27 / 32

slide-60

SLIDE 60

Piecewise projective interpolation

want to interpolate over triangles piecewise linear interpolation always works better: circumcircle preserving piecewise projective interpolation continuous across edges

meshes discretely conformally equivalent

looks better for coarse meshes (important for computer games)

Conformal Equivalence of Triangle Meshes 27 / 32

slide-61

SLIDE 61

Piecewise projective interpolation

want to interpolate over triangles piecewise linear interpolation always works better: circumcircle preserving piecewise projective interpolation continuous across edges

meshes discretely conformally equivalent

looks better for coarse meshes (important for computer games) projective

Conformal Equivalence of Triangle Meshes 27 / 32

slide-62

SLIDE 62

Piecewise projective interpolation

want to interpolate over triangles piecewise linear interpolation always works better: circumcircle preserving piecewise projective interpolation continuous across edges

meshes discretely conformally equivalent

looks better for coarse meshes (important for computer games) linear

Conformal Equivalence of Triangle Meshes 27 / 32

slide-63

SLIDE 63

1

introduction

2

smooth → discrete

3

some basics

4

discrete mapping problem

5

variational principle

6

piecewise projective interpolation

7

hyperbolic geometry

Conformal Equivalence of Triangle Meshes 28 / 32

slide-64

SLIDE 64

2D hyperbolic geometry

circumcircle induces hyperbolic metric (Klein model) → hyperbolic metric on surface vertices at infinity (cusps) conformally equivalent discrete metrics induce same hyperbolic metric no wonder about dim Tg,n log lcrij = Thurston Fock shear coordinates λij = Penner coordinates

Conformal Equivalence of Triangle Meshes 29 / 32

slide-65

SLIDE 65

2D hyperbolic geometry

circumcircle induces hyperbolic metric (Klein model) → hyperbolic metric on surface vertices at infinity (cusps) conformally equivalent discrete metrics induce same hyperbolic metric no wonder about dim Tg,n log lcrij = Thurston Fock shear coordinates λij = Penner coordinates

Conformal Equivalence of Triangle Meshes 29 / 32

slide-66

SLIDE 66

2D hyperbolic geometry

circumcircle induces hyperbolic metric (Klein model) → hyperbolic metric on surface vertices at infinity (cusps) conformally equivalent discrete metrics induce same hyperbolic metric no wonder about dim Tg,n log lcrij = Thurston Fock shear coordinates λij = Penner coordinates

Conformal Equivalence of Triangle Meshes 29 / 32

slide-67

SLIDE 67

2D hyperbolic geometry

circumcircle induces hyperbolic metric (Klein model) → hyperbolic metric on surface vertices at infinity (cusps) conformally equivalent discrete metrics induce same hyperbolic metric no wonder about dim Tg,n log lcrij = Thurston Fock shear coordinates λij = Penner coordinates

Conformal Equivalence of Triangle Meshes 29 / 32

slide-68

SLIDE 68

2D hyperbolic geometry

circumcircle induces hyperbolic metric (Klein model) → hyperbolic metric on surface vertices at infinity (cusps) conformally equivalent discrete metrics induce same hyperbolic metric no wonder about dim Tg,n log lcrij = Thurston Fock shear coordinates λij = Penner coordinates

Conformal Equivalence of Triangle Meshes 29 / 32

slide-69

SLIDE 69

2D hyperbolic geometry

circumcircle induces hyperbolic metric (Klein model) → hyperbolic metric on surface vertices at infinity (cusps) conformally equivalent discrete metrics induce same hyperbolic metric no wonder about dim Tg,n log lcrij = Thurston Fock shear coordinates λij = Penner coordinates

Conformal Equivalence of Triangle Meshes 29 / 32

slide-70

SLIDE 70

2D hyperbolic geometry

circumcircle induces hyperbolic metric (Klein model) → hyperbolic metric on surface vertices at infinity (cusps) conformally equivalent discrete metrics induce same hyperbolic metric no wonder about dim Tg,n log lcrij = Thurston Fock shear coordinates λij = Penner coordinates

log lcrij

i j

Conformal Equivalence of Triangle Meshes 29 / 32

slide-71

SLIDE 71

2D hyperbolic geometry

circumcircle induces hyperbolic metric (Klein model) → hyperbolic metric on surface vertices at infinity (cusps) conformally equivalent discrete metrics induce same hyperbolic metric no wonder about dim Tg,n log lcrij = Thurston Fock shear coordinates λij = Penner coordinates i j

Conformal Equivalence of Triangle Meshes 29 / 32

slide-72

SLIDE 72

2D hyperbolic geometry

circumcircle induces hyperbolic metric (Klein model) → hyperbolic metric on surface vertices at infinity (cusps) conformally equivalent discrete metrics induce same hyperbolic metric no wonder about dim Tg,n log lcrij = Thurston Fock shear coordinates λij = Penner coordinates i j

Conformal Equivalence of Triangle Meshes 29 / 32

slide-73

SLIDE 73

2D hyperbolic geometry

circumcircle induces hyperbolic metric (Klein model) → hyperbolic metric on surface vertices at infinity (cusps) conformally equivalent discrete metrics induce same hyperbolic metric no wonder about dim Tg,n log lcrij = Thurston Fock shear coordinates λij = Penner coordinates i j

Conformal Equivalence of Triangle Meshes 29 / 32

slide-74

SLIDE 74

2D hyperbolic geometry

circumcircle induces hyperbolic metric (Klein model) → hyperbolic metric on surface vertices at infinity (cusps) conformally equivalent discrete metrics induce same hyperbolic metric no wonder about dim Tg,n log lcrij = Thurston Fock shear coordinates λij = Penner coordinates i j

Conformal Equivalence of Triangle Meshes 29 / 32

slide-75

SLIDE 75

2D hyperbolic geometry

circumcircle induces hyperbolic metric (Klein model) → hyperbolic metric on surface vertices at infinity (cusps) conformally equivalent discrete metrics induce same hyperbolic metric no wonder about dim Tg,n log lcrij = Thurston Fock shear coordinates λij = Penner coordinates i j k l λij λik λkj λjl λli

Conformal Equivalence of Triangle Meshes 29 / 32

slide-76

SLIDE 76

3D hyperbolic geometry

Conformal Equivalence of Triangle Meshes 30 / 32

slide-77

SLIDE 77

3D hyperbolic geometry

Conformal Equivalence of Triangle Meshes 30 / 32

slide-78

SLIDE 78

3D hyperbolic geometry

Conformal Equivalence of Triangle Meshes 30 / 32

slide-79

SLIDE 79

3D hyperbolic geometry

Conformal Equivalence of Triangle Meshes 31 / 32

slide-80

SLIDE 80

3D hyperbolic geometry

Conformal Equivalence of Triangle Meshes 31 / 32

slide-81

SLIDE 81

3D hyperbolic geometry

3 1 2

Conformal Equivalence of Triangle Meshes 31 / 32

slide-82

SLIDE 82

3D hyperbolic geometry

3 1 2 λ31 λ12

Conformal Equivalence of Triangle Meshes 31 / 32

slide-83

SLIDE 83

3D hyperbolic geometry

3 1 2 λ31 λ12 λ3 λ1 λ2

Conformal Equivalence of Triangle Meshes 31 / 32

slide-84

SLIDE 84

3D hyperbolic geometry

3 1 2 λ31 λ12 λ3 λ1 λ2 ℓ31 ℓ12 ℓ23

Conformal Equivalence of Triangle Meshes 31 / 32

slide-85

SLIDE 85

3D hyperbolic geometry

3 1 2 λ31 λ12 λ3 λ1 λ2 ℓ31 ℓ12 ℓ23 ℓ12 = e

1 2(λ12−λ1−λ2)

Conformal Equivalence of Triangle Meshes 31 / 32

slide-86

SLIDE 86

3D hyperbolic geometry

3 1 2 λ31 λ12 λ3 λ1 λ2 ℓ31 ℓ12 ℓ23 ℓ12 = e

1 2(λ12+u1+u2)

ui = −λi

Conformal Equivalence of Triangle Meshes 31 / 32

slide-87

SLIDE 87

Thanks

Conformal Equivalence of Triangle Meshes 32 / 32