Discrete Conformal Maps

Transcription

1 Introduction Discrete Conformal Maps period matrices and all that Alexander Bobenko 1, Christian Mercat 2, Markus Schmies 1 1 Technische Universität Berlin (FZT 86, F5), 2 Institut de Mathématiques et de Modélisation de Montpellier (I3M, UM2) Discrete Differential Geometry 07

2 Introduction Peter Schröder & 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)

3 Introduction X. Gu & S.-T. Yau, Harvard Uni. Surface interpolation (especially staying isometric). Texture mapping. Surface matching. Remeshing, coarsening, refining.

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

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

6 Introduction Linear and quadratic conformality { az + b + o(z), f is conformal f (z) = az + b cz + d + o(z2 ). Two notions on a quad-graph : Preserve the diagonal ratio (linear), or preserve the cross-ratio (Möbius invariant). y x x y Diagonals ratio: y y x x = i ρ Crossratio: y x y x = q x y x y

7 Introduction Linear and quadratic conformality { az + b + o(z), f is conformal f (z) = az + b cz + d + o(z2 ). Two notions on a quad-graph : Preserve the diagonal ratio (linear), or preserve the cross-ratio (Möbius invariant). y x x y Diagonals ratio: y y x x = i ρ = f (y ) f (y) f (x ) f (x) Crossratio: y x y x = q x y x y

8 Introduction Linear and quadratic conformality { az + b + o(z), f is conformal f (z) = az + b cz + d + o(z2 ). Two notions on a quad-graph : Preserve the diagonal ratio (linear), or preserve the cross-ratio (Möbius invariant). y x x y Diagonals ratio: y y x x = i ρ Crossratio: y x y x = q x y x y = f (y ) f (x ) f (y) f (x) f (x ) f (y) f (x) f (y )

9 Introduction Circle patterns Circle patterns are a particular case x θ y ϕ θ ϕ x y q = e 2(θ+θ ) = e 2ϕ ρ = cos(θ θ ) cos(ϕ) sin(ϕ)

10 Introduction Hirota System A function F preserving the cross-ratio can be written in terms of 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 F (z) dz = f (x) f (y) (y x) + f (y) f (x ) (x y) + x y f (x ) f (y ) (y x ) + f (y ) f (x) (x y ) = 0 Circle patterns case: F (y) F (x) = r(x) e i θ(y) (y x).

11 Introduction Morera equation H F (z) dz = 0 Understood as Morera equations where function integration is the geometric mean for cross-ratio preserving maps g dz := g(x) g(y) (y x). (x,y) the arithmetic mean for diagonal ratio preserving maps (x,y) g dz := g(x) + g(y) 2 (y x).

12 Introduction From quadratic to linear When the quadratic case is linearized:

13 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: z 3

14 Linear theory de Rham Cohomology v n F e e v 1 v v 2 0. Vertex dual to a face. 1. Dual edges. 2. Face dual to a vertex. The double Λ = Γ Γ The chains-complex C(Λ) = C 0 (Λ) C 1 (Λ) C 2 (Λ) linear combination of vertices (0), edges (1) and faces (2). boundary operator : C k (Λ) C k 1 (Λ) Null on vertices, 2 = 0. Its kernel ker =: Z (Λ) are the closed chains or cycles. Its image are the exact chains.

15 Linear theory de Rham Cohomology The space dual to chains form the cochains, C k (Λ) := Hom(C k (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 df := f ( (x, x ) ) = f (x ) f (x), dα := α. (x,x ) A cocycle is a closed cochain α Z k (Λ). F F

16 Linear theory Metric, Hodge operator, Laplacian Scalar product weighted by ρ y x x ρ (x,x ) = i y y x x y ( (α, β) := 1 2 ρ(e) e Λ 1 e ) ( ) α β. (1) e

17 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 : α := ρ(e ) α, (2) e e C 2 (Λ) ω ω : ( ω)(x) := ω. x Verifies 2 = ( Id C k ) k. The discrete laplacian = Γ Γ := d d d d: ( (f )) (x) = V ρ(x, x k ) (f (x) f (x k )). k=1 Its kernel are the harmonic forms.

18 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 (Id i ) 2 d := π (1,0) d : C 0 (Λ) C 1 (Λ), d := d π (1,0) : C 1 (Λ) C 2 (Λ) f Ω 0 (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.

19 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, f (x) + f (y) f α := α for (x, y) 1, 2 (x,y) (x 1,x 2,x 3,x 4 ) (x 1,x 2,x 3,x 4 ) α β := (x,y) α k=1 (x k 1,x k ) (x k,x k+1 ) β f ω := f (x 1)+f (x 2 )+f (x 3 )+f (x 4 ) 4 α β, (x k+1,x k ) (x k,x k 1 ) ω (x 1,x 2,x 3,x 4 ) for (x 1, x 2, x 3, x 4 ) 2.

20 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, f (x) + f (y) f α := α for (x, y) 1, 2 (x,y) (x 1,x 2,x 3,x 4 ) (x 1,x 2,x 3,x 4 ) α β := (x,y) α k=1 (x k 1,x k ) (x k,x k+1 ) β f ω := f (x 1)+f (x 2 )+f (x 3 )+f (x 4 ) 4 α β, (x k+1,x k ) (x k,x k 1 ) ω (x 1,x 2,x 3,x 4 ) for (x 1, x 2, x 3, x 4 ) 2.

21 Linear theory External Product A form on can be averaged into a form on Λ: (x,x ) A(α ) := 1 2 (x,y) + x A(ω ) := (y,x ) (x,y ) d + (y,x ) k=1 (x k,y k,x,y k 1 ) α, (4) ω, (5) Ker(A) = Span(d ε)

22 Main Results Discrete Harmonicity Hodge Star: : C k C 2 k, (y,y ) α := ρ (x,x ) Discrete Laplacian: d := d, := d d + d d, f (x) = ρ (x,xk )(f (x) f (x k )). (x,x ) α. 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

23 Main Results Meromorphic Forms α C 1 is conformal { α = iα of 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 order 1 in v with a residue Res v α := 1 2iπ v α 0. If α is not of type (1,0) on (x, y, x, y ), it has a pole of order > 1. Discrete Riemann-Roch theorem: Existence of forms with prescribed poles and holonomies. Green Function and Potential, Cauchy Integral Formula: D f ν f (x)+f (y) x,y = 2iπ 2. Period Matrix, Jacobian, Abel s map, Riemann bilinear relations. Continuous limit theorem in locally flat regions (criticality).

24 Main Results Dirac spinor y Υ x ξ4 ξ1 ξ3 φ(y,y ) ξ2 y x (x,y,x,y ) ξ dz = 0 ˆΥ Ising model ρ(e) = sinh2k e, is critical iff the fermion ψ xy = σ x µ y 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)

25 Main Results Energies The L 2 -norm of the 1-form df is the Dirichlet energy E D (f ) := df 2 = (df, df ) = 1 ρ(x, x ) f (x ) f (x) 2 2 (x,x ) Λ 1 = E D(f Γ ) + E D (f Γ ). 2 The conformal energy of the map measures its conformality defect E C (f ) := 1 2 df i df 2.

26 Main Results Energies Dirichlet and Conformal energies are related through E C (f ) = 1 2 (df i df, df i df ) = 1 2 df i df 2 + Re(df, i df ) = df 2 + Im df df 2 = E D (f ) 2A(f ) with the algebraic area of the image A(f ) := i df df 2 2 On a face the algebraic algebra of the image reads ( ) df df = i Im (f (x ) f (x))(f (y ) f (y)) ( ) (x,y,x,y ) = 2iA f (x), f (x ), f (y), f (y )

27 Main Results Complex discrete structure (α, β) := ( (x,x ) α ) (y,y ) α α β = 1 2 := 1 cos θ e Λ 1 E D (f ) := df 2 = 1 f (x ) f (x) 2 Re (ρ e ) e Λ 1 ( ) ( sin θ 1 r (x,x ) α (y,y ) α ) r sin θ e α ( ρ e 2 β + Im (ρ e ) Re (ρ e ) e 2. ) β e ( ) ρ e 2 + Im (ρ e ) f (y ) f (y). f (x ) f (x) For x 0 Λ 0, x 0 = (y 1, y 2,..., y V ) Λ 2, (x 0, x k ) = (y k, y k+1 ) Λ 1, (f )(x 0 ) = V k=1 1 Re ( ) ( ρ e 2( f (x k ) f (x) ) +Im ( )( ρ e f (yk+1 ) f (y k ) )) ρ e

28 Main Results Algorithm Basis of holomorphic forms(a discrete Riemann surface S) find a normalized homotopy basis ℵ of (S) foreach ℵ k and ℵΓ compute ℵ Γ k k compute the real discrete harmonic form ω k on Γ s.t. γ ω k = γ ℵ Γ k compute the form ω k on Γ check it is harmonic on on Γ compute its holonomies ( ω ℵ Γ k ) k,l on the dual graph l do some linear algebra (R is a rectangular complex matrix) to get the basis of holomorphic forms (ζ k ) k = R(Id + i )(ω k ) k s.t. ( ℵ ζ Γ k ) = δ k,l l define the period matrix Π k,l := ( ℵ ζ Γ k ) l

29 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

30 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

31 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

32 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

33 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, ē}; faces faces \ {face}; lface leftface(ē); if (lface) facestoclose(lface) facestoclose(lface \ {ē}) if ( facestoclose(lface) == 0) faces faces \ {lface}; while (not(finished)); return facestoclose

34 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, ē}; faces faces \ {face}; lface leftface(ē); if (lface) facestoclose(lface) facestoclose(lface \ {ē}) if ( facestoclose(lface) == 0) faces faces \ {lface}; while (not(finished)); return facestoclose

35 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, ē}; faces faces \ {face}; lface leftface(ē); if (lface) facestoclose(lface) facestoclose(lface \ {ē}) if ( facestoclose(lface) == 0) faces faces \ {lface}; while (not(finished)); return facestoclose

36 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, ē}; faces faces \ {face}; lface leftface(ē); if (lface) facestoclose(lface) facestoclose(lface \ {ē}) if ( facestoclose(lface) == 0) faces faces \ {lface}; while (not(finished)); return facestoclose

37 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, ē}; faces faces \ {face}; lface leftface(ē); if (lface) facestoclose(lface) facestoclose(lface \ {ē}) if ( facestoclose(lface) == 0) faces faces \ {lface}; while (not(finished)); return facestoclose

38 Numerics Surfaces tiled by squares Surface Period Matrix Numerical Analysis Ω 1 = i 3 ( ) ( ) Ω2 = i 1 2i 3 1 2i 2 + 8i Ω 3 = ( ) i #vertices Ω D Ω #vertices Ω D Ω #vertices Ω D Ω

39 Numerics Wente torus Grid : Grid : Grid : Grid : τ w i exp(i ). Grid τ in τ w τ ex τ w

40 Numerics Lawson surface 1162 vertices 2498 vertices Ω l = i 3 ( ) #vertices Ω in Ω l Ω ex Ω l

41 Quasi conformal maps In the continuous case and for f (z + z 0 ) = f (z 0 ) + z ( f )(z 0 ) + z ( f )(z 0 ) + o( z ), i ( f )(z 0 ) = lim fd z, ( γ z0 2A(γ) f i )(z 0 ) = lim fdz, γ z0 2A(γ) along a loop γ around z 0. leading to the discrete definition : C 0 ( ) C 2 ( ) γ f f = [ (x, y, x, y ) i 2A(x,y,x,y ) : C 0 ( ) C 2 ( ) γ (x,y,x,y ) = (f (x ) f (x))(ȳ ȳ) ( x x)(f (y ) f (y)) (x x)(ȳ ȳ) ( x x)(y y), f f = [ (x, y, x, y ) i 2A(x,y,x,y ) (x,y,x,y ) fd Z ] fdz ]

42 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: df df = J dz dz. (x,y,x,y ) (x,y,x,y )

43 Quasi conformal maps Quasi-Conformal Maps For a discrete function, one defines the dilatation coefficient D f := f z + f z f z f z D f 1 for f z f z (quasi-conformal). Written in terms of the complex dilatation: µ f = f z f z i µ f µf

44 Quasi conformal maps Quasi-Conformal Maps For a discrete function, one defines the dilatation coefficient D f := f z + f z f z f z D f 1 for f z f z (quasi-conformal). Written in terms of the complex dilatation: µ f = f z f z i µ f µf

45 Quasi conformal maps Quasi-Conformal Maps (Arnaud Chéritat and Xavier Buff, L. Émile Picard, Toulouse)

46 Criticality Lozenges (Duffin) and Exponential u 0 = O u 1 u 2 u n = z exp(:λ: z) = k 1 + λ 2 (u k u k 1 ) 1 λ 2 (u k u k 1 ). Generalization of the well known formula exp(λ z) = ( 1 + λ z ) n + O( z2 n n ) = ( 1 + λ z 2 n 1 λ z 2 n ) n + O( z3 n 2 ).

47 Criticality Lozenges (Duffin) and Exponential The Green function on a lozenges graph is (Richard Kenyon) G(O, x) = 1 8 π 2 i C exp(:λ: x) log δ 2 λ λ dλ { log x G(O, x) = δ O,x, G(O, x) x i arg(x) on black vertices, on white vertices.

48 Criticality Lozenges (Duffin) and Exponential Differentiation of the exponential with respect to λ yields k λ k 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: f (x) + f (y) f dz := (y x). 2 (x,y) 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.

49 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 and verifies f = f dz. f (z) := 4 ( z δ 2 f dz) + λ ε, O

50 Bäcklund transformation y λ y λ x λ λ x y y x λ 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: xλ xλ yλ y λ yλ y λ x x λ y y y y x x

51 Bäcklund transformation y λ y λ x λ λ x x λ x y 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: xλ xλ yλ y λ yλ y λ x x λ y y y y x x

52 Bäcklund transformation y λ y λ x λ λ x x λ x y 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: xλ xλ yλ y λ yλ y λ x x λ y y y y x x

53 Bäcklund transformation y λ y λ x λ λ x x λ x y 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: xλ xλ yλ y λ yλ y λ x x λ y y y y x x

54 Bäcklund transformation y λ y λ x λ λ x x λ x y 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: xλ xλ yλ y λ yλ y λ x x λ y y y y x x

55 Bäcklund transformation y λ y λ x λ λ x x λ x y 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: xλ xλ yλ y λ yλ y λ x x λ y y y y x x

56 Bäcklund transformation y λ y λ x λ λ x x λ x y 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: xλ xλ yλ y λ yλ y λ x x λ y y y y x x

57 Bäcklund transformation The exponential F G = B u λ (F ) B u µ(f ) B v λ (G) exp u (: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

58 Bäcklund transformation The exponential F = B u λ (G) G = B u λ (F ) B u µ(f ) B u µ(g) B v λ (G) exp u (: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

59 Bäcklund transformation The exponential F = B u λ (G) G = B u λ (F ) B u µ(f ) B u µ(g) B v λ (G) exp u (: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

60 Bäcklund transformation The exponential F = B u λ (G) G = B u λ (F ) B u µ(f ) B u µ(g) B v λ (G) exp u (: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

61 Bäcklund transformation Solitons yλ y λ xλ x λ The vertical equation is (f λ (y) f λ (x)) (f (y) f (x)) (y x)2 = (f λ (y) f (y)) (f λ (x) f (x)) λ 2 λ x y x y With the change of variables g = ln f, the continuous limit (y = x + ɛ) is: g λ g = 2 λ sinh g λ g. 2

62 Bäcklund transformation Isomonodromic solutions, moving frame The Bäcklund transformation allows to define a notion of discrete holomorphy in Z d, for d > 1 finite, equipped with rapidities (α i ) 1 i d. x + e ρ = α i +α j α i x + e j i + e j α i α j α j x α i α j x + e i q = α2 i α 2 j f (x + e i + e j ) = L i (x + e j )f (x + e j ), L i (x; λ) = λ + α i 2α i (f (x + e i ) + f (x)) 0 λ α i The moving frame Ψ(, λ) : Z d GL 2 (C)[λ] for a prescribed Ψ(0; λ): Ψ(x + e i ; λ) = L i (x; λ)ψ(x; λ).

63 Bäcklund transformation Isomonodromic solutions, moving frame Define A( ; λ) : Z d GL 2 (C)[λ] by A(x; λ) = They satisfy the recurrent relation dψ(x; λ) Ψ 1 (x; λ). dλ A(x + e k ; λ) = dl k(x; λ) dλ L 1 k (x; λ) + L k(x; λ)a(x; λ)l 1 (x; λ). k A discrete holomorphic function f : Z d 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 orders do not depend on x Z d. Isomonodromic solutions can be constructed with prescribed boundary conditions. Example: the Green function.

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