Discrete Conformal Maps period matrices and all that Alexander - - PowerPoint PPT Presentation

Discrete Riemann Surfaces Introduction

Discrete Conformal Maps

period matrices and all that Alexander Bobenko1, Christian Mercat2, Markus Schmies1

1 Technische Universit¨ at Berlin (FZT 86, F5), 2 Institut de Math´ ematiques et de Mod´ elisation de Montpellier (I3M, UM2) Discrete Differential Geometry 07

Discrete Riemann Surfaces Introduction

Peter Schr¨

• der & al., CalTech

Global parametrization sending little circles to circles. Riemann theorem mapping a topological disk to the unit disk or with free boundaries (see M. Desbrun & al. or C. Gotsman & M. Ben-Chen)

Discrete Riemann Surfaces Introduction

• X. Gu & S.-T. Yau, Harvard Uni.

Surface interpolation (especially staying isometric). Texture

• mapping. Surface matching. Remeshing, coarsening, refining.

Discrete Riemann Surfaces Introduction

Monica K. Hurdal & al., Florida State University Commonly parametrize different surfaces to compare functions on them.

Discrete Riemann Surfaces Introduction

Tony Chan, UCLA Whether preserving little circles (see K. Stephenson) or preserving little squares.

Discrete Riemann Surfaces Introduction Linear and quadratic conformality

f is conformal ⇒ f (z) =

• az + b + o(z),

az + b cz + d + o(z2). Two notions on a “quad-graph”: – Preserve the diagonal ratio (linear), – or preserve the cross-ratio (M¨

• bius invariant).

y ′ x′ y x Diagonals ratio: y ′ − y x′ − x = i ρ y ′ − x′ x′ − y y − x x − y ′ = q Crossratio:

Discrete Riemann Surfaces Introduction Linear and quadratic conformality

f is conformal ⇒ f (z) =

• az + b + o(z),

az + b cz + d + o(z2). Two notions on a “quad-graph”: – Preserve the diagonal ratio (linear), – or preserve the cross-ratio (M¨

• bius invariant).

y ′ x′ y x Diagonals ratio: y ′ − y x′ − x = i ρ = f (y ′) − f (y) f (x′) − f (x) y ′ − x′ x′ − y y − x x − y ′ = q Crossratio:

Discrete Riemann Surfaces Introduction Linear and quadratic conformality

f is conformal ⇒ f (z) =

• az + b + o(z),

az + b cz + d + o(z2). Two notions on a “quad-graph”: – Preserve the diagonal ratio (linear), – or preserve the cross-ratio (M¨

• bius invariant).

y ′ x′ y x Diagonals ratio: y ′ − y x′ − x = i ρ y ′ − x′ x′ − y y − x x − y ′ = q Crossratio: = f (y ′) − f (x′) f (x′) − f (y) f (y) − f (x) f (x) − f (y ′)

Discrete Riemann Surfaces Introduction Circle patterns

Circle patterns are a particular case

− → x y y′ x′ θ θ′ ρ = cos(θ−θ′)−cos(ϕ)

sin(ϕ)

ϕ ϕ q = e−2(θ+θ′) = e−2ϕ

Discrete Riemann Surfaces Introduction Hirota System

A function F preserving the cross-ratio can be written in terms

• f a function f such that

F(y) − F(x) = f (x) f (y) (y − x) = “F ′(z) dz”. fulfilling on the face (x, y, x′, y′),

y ′ x′ y x

• F ′(z) dz

= f (x) f (y) (y − x) + f (y) f (x′) (x′ − y) + f (x′) f (y′) (y′ − x′) + f (y′) f (x) (x − y′) = 0 Circle patterns case: F(y) − F(x) = r(x) ei θ(y) (y − x).

Discrete Riemann Surfaces Introduction Morera equation H F ′(z) dz = 0

Understood as Morera equations where function integration is

◮ the geometric mean for cross-ratio preserving maps

• (x,y)

g dZ :=

• g(x) g(y) (y − x).

◮ the arithmetic mean for diagonal ratio preserving maps

• (x,y)

g dZ := g(x) + g(y) 2 (y − x).

Discrete Riemann Surfaces Introduction From quadratic to linear

When the quadratic case is linearized:

Discrete Riemann Surfaces Introduction From quadratic to linear

Circle patterns preserve circles and intersection angles. Linear maps preserve the shape of dual/primal polygons, the derivative f ′(z) locally inflates and turns each polygon, the Morera equation

• f ′(z) dZ = 0 insures that they fit together. Compare with the

continuous case:

http://ens.math.univ-montp2.fr/SPIP/-Deformer-par-une-application-

z → z3

Discrete Riemann Surfaces Linear theory de Rham Cohomology

• 2. Face dual to a vertex.
• 1. Dual edges.
• 0. Vertex dual to a face.

e∗ F ∗ e v1 v2 v vn

The double Λ = Γ ⊕ Γ∗ The chains-complex C(Λ) = C0(Λ) ⊕ C1(Λ) ⊕ C2(Λ) linear combination of vertices (0), edges (1) and faces (2). boundary operator ∂ : Ck(Λ) → Ck−1(Λ) Null on vertices, ∂2 = 0. Its kernel ker ∂ =: Z•(Λ) are the closed chains or cycles. Its image are the exact chains.

Discrete Riemann Surfaces Linear theory de Rham Cohomology

The space dual to chains form the cochains, C k(Λ) := Hom(Ck(Λ), C). Evaluation is denoted f (x),

• (x,x′) α,
• F ω.

The coboundary is dual to the boundary. d : C k(Λ) → C k+1(Λ), defined by Stokes formula

• (x,x′)

df := f

• ∂(x, x′)
• = f (x′) − f (x),
• F

dα :=

• ∂F

α. A cocycle is a closed cochain α ∈ Z k(Λ).

Discrete Riemann Surfaces Linear theory Metric, Hodge operator, Laplacian

Scalar product weighted by ρ

ρ(x,x′) = −i y′−y

x′−x

y ′ x x′ y

(α, β) := 1

2

• e∈Λ1

ρ(e)

• e

α

e

¯ β

• .

(1)

Discrete Riemann Surfaces Linear theory Metric, Hodge operator, Laplacian

A Hodge operator ∗ ∗ : C k(Λ) → C 2−k(Λ) C 0(Λ) ∋ f → ∗f :

• F

∗f := f (F ∗), C 1(Λ) ∋ α → ∗α :

• e

∗α := −ρ(e∗)

• e∗ α,

(2) C 2(Λ) ∋ ω → ∗ω : (∗ω)(x) :=

• x∗ ω.

Verifies ∗2 = (−IdC k)k. The discrete laplacian ∆ = ∆Γ ⊕ ∆Γ∗ := −d ∗ d ∗ − ∗ d ∗ d: (∆(f )) (x) =

V

• k=1

ρ(x, xk) (f (x) − f (xk)) . Its kernel are the harmonic forms.

Discrete Riemann Surfaces Linear theory Holomorphic forms

α ∈ C 1(Λ) is conformal iff dα = 0 and ∗ α = −iα, (3) π(1,0) = 1 2(Id + i ∗) π(0,1) = 1 2(Id − i ∗) d′ := π(1,0)◦d : C 0(Λ) → C 1(Λ), d′ := d◦π(1,0) : C 1(Λ) → C 2(Λ) f ∈ Ω0(Λ) iff d′(f ) = 0. − ∗ d ∗ = d∗ the adjoint of the coboundary C k(Λ) = Im d ⊕⊥ Im d∗ ⊕⊥ Ker ∆, Ker ∆ = Ker d ∩ Ker d∗ = Ker d′ ⊕⊥ Ker d′′.

Discrete Riemann Surfaces Linear theory External Product

∧ : C k(♦) × C l(♦) → C k+l(♦) s.t. (α, β) =

• α ∧ ∗¯

β For f , g ∈ C 0(♦), α, β ∈ C 1(♦) et ω ∈ C 2(♦): (f · g)(x) :=f (x) · g(x) for x ∈ ♦0,

• (x,y)

f · α :=f (x) + f (y) 2

• (x,y)

α for (x, y) ∈ ♦1,

• (x1,x2,x3,x4)

α ∧ β :=1

4 4

• k=1
• (xk−1,xk)

α

• (xk,xk+1)

β −

• (xk+1,xk)

α

• (xk,xk−1)

β,

• (x1,x2,x3,x4)

f · ω :=

f (x1)+f (x2)+f (x3)+f (x4)

4

• (x1,x2,x3,x4)

ω for (x1, x2, x3, x4) ∈ ♦2.

Discrete Riemann Surfaces Linear theory External Product

∧ : C k(♦) × C l(♦) → C k+l(♦) s.t. (α, β) =

• α ∧ ∗¯

β For f , g ∈ C 0(♦), α, β ∈ C 1(♦) et ω ∈ C 2(♦): (f · g)(x) :=f (x) · g(x) for x ∈ ♦0,

• (x,y)

f · α :=f (x) + f (y) 2

• (x,y)

α for (x, y) ∈ ♦1,

• (x1,x2,x3,x4)

α ∧ β :=1

4 4

• k=1
• (xk−1,xk)

α

• (xk,xk+1)

β −

• (xk+1,xk)

α

• (xk,xk−1)

β,

• (x1,x2,x3,x4)

f · ω :=

f (x1)+f (x2)+f (x3)+f (x4)

4

• (x1,x2,x3,x4)

ω for (x1, x2, x3, x4) ∈ ♦2.

Discrete Riemann Surfaces Linear theory External Product

A form on ♦ can be averaged into a form on Λ:

• (x,x′)

A(α♦) := 1

2

  

• (x,y)

+

• (y,x′)

+

• (x,y′)

+

• (y′,x′)

   α♦, (4)

• x∗

A(ω♦) := 1

2 d

• k=1
• (xk,yk,x,yk−1)

ω♦, (5) Ker(A) = Span(d♦ε)

Discrete Riemann Surfaces Main Results Discrete Harmonicity

◮ Hodge Star:

∗ : C k → C 2−k,

• (y,y′) ∗ α := ρ(x,x′)
• (x,x′) α.

◮ Discrete Laplacian:

d∗ := − ∗ d ∗, ∆ := d d∗ + d∗d, ∆ f (x) = ρ(x,xk)(f (x) − f (xk)).

◮ Hodge Decomposition:

C k = Imd ⊕⊥ Imd∗ ⊕⊥ Ker ∆, Ker ∆1 = C (1,0) ⊕⊥ C (0,1).

◮ Weyl Lemma:

∆ f = 0 ⇐ ⇒

• f ∗ ∆g = 0,

∀g compact.

◮ Green Identity:

• D(f ∗ ∆g − g ∗ ∆f ) =
• ∂D(f ∗ dg − g ∗ df ).

cf Wardetzky, Polthier, Glickenstein, Novikov, Wilson

Discrete Riemann Surfaces Main Results Meromorphic Forms

α ∈ C 1 is conformal ⇐ ⇒

• ∗α = −iα
• f type (1,0)

dα = 0 closed.

◮ α of type (1,0) (i.e.

• (y,y′) α = iρ(x,x′)
• (x,x′) α) has a pole of
• rder 1 in v with a residue

Resv α :=

1 2iπ

• ∂v∗ α = 0.

◮ If α is not of type (1,0) on (x, y, x′, y′), it has a pole of

• rder > 1.

◮ Discrete Riemann-Roch theorem: Existence of forms with

prescribed poles and holonomies.

◮ Green Function and Potential, Cauchy Integral Formula:

• ∂D f · νx,y = 2iπ f (x)+f (y)

2

.

◮ Period Matrix, Jacobian, Abel’s map, Riemann bilinear

relations.

◮ Continuous limit theorem in locally flat regions (criticality).

Discrete Riemann Surfaces Main Results Dirac spinor ξ2 ξ3 ξ4 y ξ1 φ(y,y′) x′ y ′ x

• (x,y,x′,y′)

ξ √ dZ = 0

ˆ Υ ˜ Υ

Ising model ρ(e) = sinh2Ke, is critical iff the fermion ψxy = σx µy is a discrete massless Dirac spinor. Criticality has a meaning at finite size: compatibility with holomorphicity. Off criticality: massive spinor. (see V. Bazhanov, D. Cimasoni and U. Pinkall)

Discrete Riemann Surfaces Main Results Energies

The L2-norm of the 1-form df is the Dirichlet energy ED(f ) := df 2 = (df , df ) = 1 2

• (x,x′)∈Λ1

ρ(x, x′)

• f (x′) − f (x)
• 2

= ED(f |Γ) + ED(f |Γ∗) 2 . The conformal energy of the map measures its conformality defect EC(f ) := 1

2df − i ∗ df 2.

Discrete Riemann Surfaces Main Results Energies

Dirichlet and Conformal energies are related through EC(f ) = 1

2 (df − i ∗ df , df − i ∗ df )

= 1

2df 2 + 1 2−i ∗ df 2 + Re(df , −i ∗ df )

= df 2 + Im

• ♦2

df ∧ df = ED(f ) − 2A(f ) with the algebraic area of the image A(f ) := i 2

• ♦2

df ∧ df On a face the algebraic algebra of the image reads

• (x,y,x′,y′)

df ∧ df = i Im

• (f (x′) − f (x))(f (y′) − f (y))
• = −2iA
• f (x), f (x′), f (y), f (y′)

Discrete Riemann Surfaces Main Results Complex discrete structure

• (x,x′) ∗α
• (y,y′) ∗α
• :=

1 cos θ − sin θ −1

r

r sin θ

(x,x′) α

• (y,y′) α
• .

(α, β) :=

• ♦

α ∧ ∗¯ β = 1

2

• e∈Λ1
• e α

Re (ρe)

• |ρe|2
• e

¯ β + Im (ρe)

• e∗

¯ β

• ED(f ) := df 2 = 1

2

• e∈Λ1

|f (x′) − f (x)|2 Re (ρe)

• |ρe|2 + Im (ρe) f (y′) − f (y)

f (x′) − f (x)

• .

For x0 ∈ Λ0, x∗

0 = (y1, y2, . . . , yV ) ∈ Λ2, (x0, xk)∗ = (yk, yk+1) ∈ Λ1,

∆(f )(x0) =

V

• k=1

1 Re

• ρe
• |ρe|2

f (xk)−f (x)

• +Im
• ρe
• f (yk+1)−f (yk)

Discrete Riemann Surfaces Main Results Algorithm

◮ Basis of holomorphic forms(a discrete Riemann surface S) ◮ find a normalized homotopy basis ℵ of (S) ◮ foreach ℵk ◮

◮ compute ℵΓ

k and ℵΓ∗ k

◮ compute the real discrete harmonic form ωk on Γ s.t.

• γ ωk = γ · ℵΓ

k

◮ compute the form ∗ωk on Γ∗ ◮ check it is harmonic on on Γ∗ ◮ compute its holonomies (

• ℵΓ∗

ℓ ∗ωk)k,ℓ on the dual graph

◮ do some linear algebra (R is a rectangular complex matrix) to

get the basis of holomorphic forms (ζk)k = R(Id + i ∗)(ωk)k s.t. (

• ℵΓ

ℓ ζk) = δk,ℓ

◮ define the period matrix Πk,ℓ := (

• ℵΓ∗

ℓ ζk)

Discrete Riemann Surfaces Main Results Algorithm

◮ spanning tree(a root vertex) ◮ tree ← sd ← {root}; sdp1 ← ∅; points ← all the vertices; ◮ while points = ∅ ◮

points ← points \ sd

◮

foreach v ∈ sd

◮

◮ foreach v ′ ∼ v ◮ ◮ if v ′ ∈ points ◮

sdp1 ← sdp1 ∪ {v ′}

◮

tree ← tree ∪ {(v, v ′)}

◮

sd ← sdp1; sdp1 ← ∅

◮ return tree

Discrete Riemann Surfaces Main Results Algorithm

◮ spanning tree(a root vertex) ◮ tree ← sd ← {root}; sdp1 ← ∅; points ← all the vertices; ◮ while points = ∅ ◮

points ← points \ sd

◮

foreach v ∈ sd

◮

◮ foreach v ′ ∼ v ◮ ◮ if v ′ ∈ points ◮

sdp1 ← sdp1 ∪ {v ′}

◮

tree ← tree ∪ {(v, v ′)}

◮

sd ← sdp1; sdp1 ← ∅

◮ return tree

Discrete Riemann Surfaces Main Results Algorithm

◮ spanning tree(a root vertex) ◮ tree ← sd ← {root}; sdp1 ← ∅; points ← all the vertices; ◮ while points = ∅ ◮

points ← points \ sd

◮

foreach v ∈ sd

◮

◮ foreach v ′ ∼ v ◮ ◮ if v ′ ∈ points ◮

sdp1 ← sdp1 ∪ {v ′}

◮

tree ← tree ∪ {(v, v ′)}

◮

sd ← sdp1; sdp1 ← ∅

◮ return tree

Discrete Riemann Surfaces Main Results Algorithm

◮ spanning tree(a root vertex) ◮ tree ← sd ← {root}; sdp1 ← ∅; points ← all the vertices; ◮ while points = ∅ ◮

points ← points \ sd

◮

foreach v ∈ sd

◮

◮ foreach v ′ ∼ v ◮ ◮ if v ′ ∈ points ◮

sdp1 ← sdp1 ∪ {v ′}

◮

tree ← tree ∪ {(v, v ′)}

◮

sd ← sdp1; sdp1 ← ∅

◮ return tree

Discrete Riemann Surfaces Main Results Algorithm

◮ fundamental polygon(a spanning tree) ◮ faces ← all the faces; edges ← tree ∪ tree;

toClose : faces → P(all the edges);

◮ foreach (face ∈ faces) facesToClose(face) ← ∂(face) \ edges ◮ do ◮

◮ finished ← true ◮ foreach (face ∈ faces such that |facesToClose(face)| == 1) ◮ ◮ finished ← false; e = facesToClose(face); ◮ edges ← edges \ {e, ¯

e}; faces ← faces \ {face}; lface ← leftFace(¯ e);

◮ if (lface) ◮

facesToClose(lface) ← facesToClose(lface \ {¯ e})

◮

if (|facesToClose(lface)| == 0) faces ← faces \ {lface};

◮ while (not(finished)); ◮ return facesToClose

Discrete Riemann Surfaces Main Results Algorithm

◮ fundamental polygon(a spanning tree) ◮ faces ← all the faces; edges ← tree ∪ tree;

toClose : faces → P(all the edges);

◮ foreach (face ∈ faces) facesToClose(face) ← ∂(face) \ edges ◮ do ◮

◮ finished ← true ◮ foreach (face ∈ faces such that |facesToClose(face)| == 1) ◮ ◮ finished ← false; e = facesToClose(face); ◮ edges ← edges \ {e, ¯

e}; faces ← faces \ {face}; lface ← leftFace(¯ e);

◮ if (lface) ◮

facesToClose(lface) ← facesToClose(lface \ {¯ e})

◮

if (|facesToClose(lface)| == 0) faces ← faces \ {lface};

◮ while (not(finished)); ◮ return facesToClose

Discrete Riemann Surfaces Main Results Algorithm

◮ fundamental polygon(a spanning tree) ◮ faces ← all the faces; edges ← tree ∪ tree;

toClose : faces → P(all the edges);

◮ foreach (face ∈ faces) facesToClose(face) ← ∂(face) \ edges ◮ do ◮

◮ finished ← true ◮ foreach (face ∈ faces such that |facesToClose(face)| == 1) ◮ ◮ finished ← false; e = facesToClose(face); ◮ edges ← edges \ {e, ¯

e}; faces ← faces \ {face}; lface ← leftFace(¯ e);

◮ if (lface) ◮

facesToClose(lface) ← facesToClose(lface \ {¯ e})

◮

if (|facesToClose(lface)| == 0) faces ← faces \ {lface};

◮ while (not(finished)); ◮ return facesToClose

Discrete Riemann Surfaces Main Results Algorithm

◮ fundamental polygon(a spanning tree) ◮ faces ← all the faces; edges ← tree ∪ tree;

toClose : faces → P(all the edges);

◮ foreach (face ∈ faces) facesToClose(face) ← ∂(face) \ edges ◮ do ◮

◮ finished ← true ◮ foreach (face ∈ faces such that |facesToClose(face)| == 1) ◮ ◮ finished ← false; e = facesToClose(face); ◮ edges ← edges \ {e, ¯

e}; faces ← faces \ {face}; lface ← leftFace(¯ e);

◮ if (lface) ◮

facesToClose(lface) ← facesToClose(lface \ {¯ e})

◮

if (|facesToClose(lface)| == 0) faces ← faces \ {lface};

◮ while (not(finished)); ◮ return facesToClose

Discrete Riemann Surfaces Main Results Algorithm

◮ fundamental polygon(a spanning tree) ◮ faces ← all the faces; edges ← tree ∪ tree;

toClose : faces → P(all the edges);

◮ foreach (face ∈ faces) facesToClose(face) ← ∂(face) \ edges ◮ do ◮

◮ finished ← true ◮ foreach (face ∈ faces such that |facesToClose(face)| == 1) ◮ ◮ finished ← false; e = facesToClose(face); ◮ edges ← edges \ {e, ¯

e}; faces ← faces \ {face}; lface ← leftFace(¯ e);

◮ if (lface) ◮

facesToClose(lface) ← facesToClose(lface \ {¯ e})

◮

if (|facesToClose(lface)| == 0) faces ← faces \ {lface};

◮ while (not(finished)); ◮ return facesToClose

Discrete Riemann Surfaces Numerics Surfaces tiled by squares

Surface Period Matrix Numerical Analysis

Ω1 = i

3

• 5

−4 −4 5

• #vertices

ΩD − Ω1∞ 25 1.13 · 10

−8

106 3.38 · 10

−8

430 4.75 · 10

−8

1726 1.42 · 10

−7

6928 1.35 · 10

−6

Ω2 = 1

3

−2 + √ 8i 1 − √ 2i 1 − √ 2i −2 + √ 8i

• #vertices

ΩD − Ω2∞ 14 3.40 · 10

−2

62 9.51 · 10

−3

254 2.44 · 10

−3

1022 6.12 · 10

−4

4096 1.53 · 10

−4

Ω3 =

i √ 3

• 2

−1 −1 2

• #vertices

ΩD − Ω3∞ 22 3.40 · 10

−3

94 9.51 · 10

−3

382 2.44 · 10

−4

1534 6.12 · 10

−5

6142 1.53 · 10

−6

Discrete Riemann Surfaces Numerics Wente torus

Grid : 10 × 10 Grid : 20 × 20 Grid : 40 × 40 Grid : 80 × 80

τw ≈ 0.41300 . . . + 0.91073 . . . i ≈ exp(i1.145045 . . . .).

Grid τin − τw τex − τw 10 × 10 1.24 · 10−3 2.58 · 10−4 20 × 20 2.10 · 10−4 5.88 · 10−4 40 × 40 3.87 · 10−5 8.49 · 10−5 80 × 80 6.54 · 10−6 7.32 · 10−5

Discrete Riemann Surfaces Numerics Lawson surface

1162 vertices 2498 vertices

Ωl = i √ 3

• 2

−1 −1 2

• #vertices

Ωin − Ωl∞ Ωex − Ωl∞ 1162 1.68 · 10−3 1.68 · 10−3 2498 3.01 · 10−3 3.20 · 10−3 10090 8.55 · 10−3 8.56 · 10−3

Discrete Riemann Surfaces Quasi conformal maps

∂ and ¯ ∂ In the continuous case for f (z + z0) = f (z0) + z × (∂f )(z0) + ¯ z × (¯ ∂f )(z0) + o(|z|), (∂f )(z0) = lim

γ→z0

i 2A(γ)

• γ

fd¯ z, (¯ ∂f )(z0) = − lim

γ→z0

i 2A(γ)

• γ

fdZ, along a loop γ around z0. leading to the discrete definition ∂ : C 0(♦) → C 2(♦) f → ∂f =

• (x, y, x′, y′)

→ −

i 2A(x,y,x′,y′)

• (x,y,x′,y′)

fd ¯ Z

• = (f (x′)−f (x))(¯

y′−¯ y)−(¯ x′−¯ x)(f (y′)−f (y)) (x′−x)(¯ y′−¯ y)−(¯ x′−¯ x)(y′−y)

, ¯ ∂ : C 0(♦) → C 2(♦) f → ¯ ∂f =

• (x, y, x′, y′)

→ −

i 2A(x,y,x′,y′)

• (x,y,x′,y′)

fdZ

Discrete Riemann Surfaces Quasi conformal maps

A conformal map f fulfills ¯ ∂f ≡ 0 and (with Z(u) denoted u) ∂f (x, y, x′, y′) = f (y′) − f (y) y′ − y = f (x′) − f (x) x′ − x . The jacobian J = |∂f |2 − |¯ ∂f |2 compares the areas:

• (x,y,x′,y′)

df ∧ df = J

• (x,y,x′,y′)

dZ ∧ dZ.

Discrete Riemann Surfaces Quasi conformal maps Quasi-Conformal Maps

For a discrete function, one defines the dilatation coefficient Df := |fz| + |f¯

z|

|fz| − |f¯

z|

Df ≥ 1 for |f¯

z| ≤ |fz| (quasi-conformal). Written in terms of the

complex dilatation: µf = f¯

z

fz

√µf → i√µf

Discrete Riemann Surfaces Quasi conformal maps Quasi-Conformal Maps

For a discrete function, one defines the dilatation coefficient Df := |fz| + |f¯

z|

|fz| − |f¯

z|

Df ≥ 1 for |f¯

z| ≤ |fz| (quasi-conformal). Written in terms of the

complex dilatation: µf = f¯

z

fz

→ i√µf √µf

Discrete Riemann Surfaces Quasi conformal maps Quasi-Conformal Maps

(Arnaud Ch´ eritat and Xavier Buff, L. ´ Emile Picard, Toulouse)

Discrete Riemann Surfaces Criticality Lozenges (Duffin) and Exponential

u1 u2 un = z u0 = O

exp(:λ: z) =

• k

1 + λ

2(uk − uk−1)

1 − λ

2(uk − uk−1).

Generalization of the well known formula exp(λ z) =

• 1 + λ z

n n + O(z2 n ) =

• 1 + λ z

2 n

1 − λ z

2 n

n + O(z3 n2 ).

Discrete Riemann Surfaces Criticality Lozenges (Duffin) and Exponential

The Green function on a lozenges graph is (Richard Kenyon) G(O, x) = − 1 8 π2 i

• C

exp(:λ: x) log δ

2λ

λ dλ ∆G(O, x) = δO,x, G(O, x)∼x→∞

• log |x|
• n black vertices,

i arg(x)

• n white vertices.

Discrete Riemann Surfaces Criticality Lozenges (Duffin) and Exponential

Differentiation of the exponential with respect to λ yields ∂k ∂λk exp(:λ: z) =: Z :k: exp(:λ: z) and λ = 0 defines the discrete polynomials Z :k: fulfilling exp(:λ: z) = Z :k:

k! , absolutely convergent for λ · δ < 1.

The primitive of a conformal map is itself conformal:

• (x,y)

f dZ := f (x) + f (y) 2 (y − x). One can solve “differential equations”, and polynomials fulfill Z :k+1: = (k + 1)

• Z :k: dZ,

while the exponential fulfills exp(:λ: z) = 1 λ

• exp(:λ: z) dZ.

Discrete Riemann Surfaces Criticality Lozenges (Duffin) and Exponential

There exists a discrete duality, with ε = ±1 on black and white vertices, f † = ε ¯ f is conformal. The derivative f ′ of a conformal map f is f ′(z) := 4 δ2 z

O

f †dZ † + λ ε, and verifies f =

• f ′dZ.

Discrete Riemann Surfaces B¨ acklund transformation

x′ x′

λ

y ′

λ

x xλ y y ′ yλ λ y x′ x y ′

We impose “vertically” the same equation as “horizontally”. The 3D consistency (see Y. Suris) or Yang-Baxter equation yields integrability for rhombic quad-graphs:

y x′ y′ y′

λ

xλ yλ x′

λ

≡ y x′ y′ y′

λ

xλ yλ x

Discrete Riemann Surfaces B¨ acklund transformation

x′ x′

λ

y ′

λ

x xλ y y ′ yλ y x′ x λ

We impose “vertically” the same equation as “horizontally”. The 3D consistency (see Y. Suris) or Yang-Baxter equation yields integrability for rhombic quad-graphs:

y x′ y′ y′

λ

xλ yλ x′

λ

≡ y x′ y′ y′

λ

xλ yλ x

Discrete Riemann Surfaces B¨ acklund transformation

x′ x′

λ

y ′

λ

x xλ y y ′ yλ λ

We impose “vertically” the same equation as “horizontally”. The 3D consistency (see Y. Suris) or Yang-Baxter equation yields integrability for rhombic quad-graphs:

y x′ y′ y′

λ

xλ yλ x′

λ

≡ y x′ y′ y′

λ

xλ yλ x

Discrete Riemann Surfaces B¨ acklund transformation

x′ x′

λ

y ′

λ

x xλ y y ′ yλ λ

We impose “vertically” the same equation as “horizontally”. The 3D consistency (see Y. Suris) or Yang-Baxter equation yields integrability for rhombic quad-graphs:

y x′ y′ y′

λ

xλ yλ x′

λ

≡ y x′ y′ y′

λ

xλ yλ x

Discrete Riemann Surfaces B¨ acklund transformation

x′ x′

λ

y ′

λ

x xλ y y ′ yλ λ

We impose “vertically” the same equation as “horizontally”. The 3D consistency (see Y. Suris) or Yang-Baxter equation yields integrability for rhombic quad-graphs:

y x′ y′ y′

λ

xλ yλ x′

λ

≡ y x′ y′ y′

λ

xλ yλ x

Discrete Riemann Surfaces B¨ acklund transformation

x′ x′

λ

y ′

λ

x xλ y y ′ yλ λ

We impose “vertically” the same equation as “horizontally”. The 3D consistency (see Y. Suris) or Yang-Baxter equation yields integrability for rhombic quad-graphs:

y x′ y′ y′

λ

xλ yλ x′

λ

≡ y x′ y′ y′

λ

xλ yλ x

Discrete Riemann Surfaces B¨ acklund transformation

x′ x′

λ

y ′

λ

x xλ y y ′ yλ λ

We impose “vertically” the same equation as “horizontally”. The 3D consistency (see Y. Suris) or Yang-Baxter equation yields integrability for rhombic quad-graphs:

y x′ y′ y′

λ

xλ yλ x′

λ

≡ y x′ y′ y′

λ

xλ yλ x

Discrete Riemann Surfaces B¨ acklund transformation The exponential

Bu

µ(F)

F G = Bu

λ(F)

Bv

−λ(G)

expu(:λ:F) := ∂

∂v Bv −λ(G)|v=u

A family of CR-preserving maps, initial condition u, “vertical” parameter λ. Induces a linear map between the tangent spaces. Its kernel: discrete exponential exp(:λ: F). Form a basis

Discrete Riemann Surfaces B¨ acklund transformation The exponential

Bu

µ(F)

Bu

−µ(G)

F = Bu

−λ(G)

G = Bu

λ(F)

Bv

−λ(G)

expu(:λ:F) := ∂

∂v Bv −λ(G)|v=u

A family of CR-preserving maps, initial condition u, “vertical” parameter λ. Induces a linear map between the tangent spaces. Its kernel: discrete exponential exp(:λ: F). Form a basis

Discrete Riemann Surfaces B¨ acklund transformation The exponential

Bu

µ(F)

Bu

−µ(G)

F = Bu

−λ(G)

G = Bu

λ(F)

Bv

−λ(G)

expu(:λ:F) := ∂

∂v Bv −λ(G)|v=u

A family of CR-preserving maps, initial condition u, “vertical” parameter λ. Induces a linear map between the tangent spaces. Its kernel: discrete exponential exp(:λ: F). Form a basis

Discrete Riemann Surfaces B¨ acklund transformation The exponential

Bu

µ(F)

Bu

−µ(G)

expu(:λ:F) := ∂

∂v Bv −λ(G)|v=u

Bv

−λ(G)

F = Bu

−λ(G)

G = Bu

λ(F)

A family of CR-preserving maps, initial condition u, “vertical” parameter λ. Induces a linear map between the tangent spaces. Its kernel: discrete exponential exp(:λ: F). Form a basis

Discrete Riemann Surfaces B¨ acklund transformation Solitons

The vertical equation is (fλ(y) − fλ(x)) (f (y) − f (x)) (fλ(y) − f (y)) (fλ(x) − f (x)) = (y − x)2 λ2

x′ x′

λ

y ′

λ

x xλ y y ′ yλ λ

With the change of variables g = ln f , the continuous limit (y = x + ) is:

• g′

λ × g′ = 2

λ sinh gλ − g 2 .

Discrete Riemann Surfaces B¨ acklund transformation Isomonodromic solutions, moving frame

The B¨ acklund transformation allows to define a notion of discrete holomorphy in Zd, for d > 1 finite, equipped with rapidities (αi)1≤i≤d.

q =

α2

i

α2

j

ρ = αi+αj

αi−αj

f (x + ei + ej) = Li(x + ej)f (x + ej), x x x + ei x + ei + ej x + ej αi αj αj αi

Li(x; λ) =  λ + αi −2αi(f (x + ei) + f (x)) λ − αi   The moving frame Ψ(·, λ) : Zd → GL2(C)[λ] for a prescribed Ψ(0; λ): Ψ(x + ei; λ) = Li(x; λ)Ψ(x; λ).

Discrete Riemann Surfaces B¨ acklund transformation Isomonodromic solutions, moving frame

Define A(·; λ) : Zd → GL2(C)[λ] by A(x; λ) = dΨ(x; λ) dλ Ψ−1(x; λ). They satisfy the recurrent relation A(x + ek; λ) = dLk(x; λ) dλ L−1

k (x; λ) + Lk(x; λ)A(x; λ)L−1 k (x; λ).

A discrete holomorphic function f : Zd → C is called isomonodromic, if, for some choice of A(0; λ), the matrices A(x; λ) are meromorphic in λ, with poles whose positions and

• rders do not depend on x ∈ Zd. Isomonodromic solutions can be

constructed with prescribed boundary conditions. Example: the Green function.

Discrete Riemann Surfaces Inter2Geo eContent+ European project

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