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 Bobenko 1 , Christian Mercat 2 , Markus Schmies 1 1 Technische Universit¨ at Berlin (FZT 86, F5), 2 Institut de Math´ ematiques et de Mod´ elisation de Montpellier (I3M, UM2) D iscrete D ifferential G eometry 07

Discrete Riemann Surfaces Introduction Peter Schr¨ oder & 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 � az + b + o ( z ) , f is conformal ⇒ f ( z ) = az + b cz + d + o ( z 2 ) . Two notions on a “quad-graph”: – Preserve the diagonal ratio (linear), – or preserve the cross-ratio (M¨ obius invariant). Diagonals ratio: y ′ y ′ − y x ′ x ′ − x = i ρ x Crossratio: y ′ − x ′ y − x y x − y ′ = q x ′ − y

Discrete Riemann Surfaces Introduction Linear and quadratic conformality � az + b + o ( z ) , f is conformal ⇒ f ( z ) = az + b cz + d + o ( z 2 ) . Two notions on a “quad-graph”: – Preserve the diagonal ratio (linear), – or preserve the cross-ratio (M¨ obius invariant). Diagonals ratio: y ′ y ′ − y x ′ − x = i ρ = f ( y ′ ) − f ( y ) x ′ f ( x ′ ) − f ( x ) x Crossratio: y ′ − x ′ y − x y x − y ′ = q x ′ − y

Discrete Riemann Surfaces Introduction Linear and quadratic conformality � az + b + o ( z ) , f is conformal ⇒ f ( z ) = az + b cz + d + o ( z 2 ) . Two notions on a “quad-graph”: – Preserve the diagonal ratio (linear), – or preserve the cross-ratio (M¨ obius invariant). Diagonals ratio: y ′ y ′ − y x ′ x ′ − x = i ρ x Crossratio: y ′ − x ′ y − x y x − y ′ = q x ′ − y = f ( y ′ ) − f ( x ′ ) f ( y ) − f ( x ) f ( x ′ ) − f ( y ) f ( x ) − f ( y ′ )

Discrete Riemann Surfaces Introduction Circle patterns Circle patterns are a particular case − → y ′ q = e − 2( θ + θ ′ ) ϕ = e − 2 ϕ θ ′ θ x ′ x ϕ ρ = cos( θ − θ ′ ) − cos( ϕ ) sin( ϕ ) y

Discrete Riemann Surfaces 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 ” . y ′ fulfilling on the face ( x , y , x ′ , y ′ ), x ′ x y � f ( x ) f ( y ) ( y − x ) + f ( y ) f ( x ′ ) ( x ′ − y ) + F ′ ( z ) dz = 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 ) .

Discrete Riemann Surfaces Introduction H F ′ ( z ) dz = 0 Morera equation 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 � g dZ := g ( x ) + g ( y ) ( y − x ) . 2 ( x , y )

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 �→ z 3

Discrete Riemann Surfaces Linear theory de Rham Cohomology v n v 1 e ∗ F ∗ v e 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.

Discrete Riemann Surfaces 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 ) , F ω . ( x , x ′ ) α, The coboundary is dual to the boundary. d : C k (Λ) → C k +1 (Λ), defined by Stokes formula � �� � ∂ ( x , x ′ ) = f ( x ′ ) − f ( x ) , � � df := f d α := α. F ∂ F ( x , x ′ ) A cocycle is a closed cochain α ∈ Z k (Λ).

Discrete Riemann Surfaces Linear theory Metric, Hodge operator, Laplacian Scalar product weighted by ρ y ′ ρ ( x , x ′ ) = − i y ′ − y x ′ − x x ′ x y �� � �� � � ¯ ( α, β ) := 1 ρ ( e ) α β . (1) 2 e e e ∈ Λ 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 := − ρ ( e ∗ ) �→ ∗ α : e ∗ α, (2) e �� C 2 (Λ) ∋ ω �→ ∗ ω : ( ∗ ω )( x ) := x ∗ ω. Verifies ∗ 2 = ( − Id C k ) k . The discrete laplacian ∆ = ∆ Γ ⊕ ∆ Γ ∗ := − d ∗ d ∗ − ∗ d ∗ d : V � ρ ( x , x k ) ( f ( x ) − f ( x k )) . (∆( f )) ( x ) = k =1 Its kernel are the harmonic forms .

Discrete Riemann Surfaces Linear theory Holomorphic forms α ∈ C 1 (Λ) is conformal iff d α = 0 and ∗ α = − i α, (3) π (1 , 0) = 1 π (0 , 1) = 1 2(Id + i ∗ ) 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 ∈ 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 ′′ .

Discrete Riemann Surfaces Linear theory External Product �� α ∧ ∗ ¯ ∧ : C k ( ♦ ) × C l ( ♦ ) → C k + l ( ♦ ) s.t. ( α, β ) = 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 · α := f ( x ) + f ( y ) � � α for ( x , y ) ∈ ♦ 1 , 2 ( x , y ) ( x , y ) 4 �� � � � � � α ∧ β := 1 α β − α β, 4 k =1 ( x 1 , x 2 , x 3 , x 4 ) ( x k − 1 , x k ) ( x k , x k +1 ) ( x k +1 , x k ) ( x k , x k − 1 ) �� �� f ( x 1 )+ f ( x 2 )+ f ( x 3 )+ f ( x 4 ) f · ω := ω 4 ( x 1 , x 2 , x 3 , x 4 ) ( x 1 , x 2 , x 3 , x 4 ) for ( x 1 , x 2 , x 3 , x 4 ) ∈ ♦ 2 .

Discrete Riemann Surfaces Linear theory External Product �� α ∧ ∗ ¯ ∧ : C k ( ♦ ) × C l ( ♦ ) → C k + l ( ♦ ) s.t. ( α, β ) = 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 · α := f ( x ) + f ( y ) � � α for ( x , y ) ∈ ♦ 1 , 2 ( x , y ) ( x , y ) 4 �� � � � � � α ∧ β := 1 α β − α β, 4 k =1 ( x 1 , x 2 , x 3 , x 4 ) ( x k − 1 , x k ) ( x k , x k +1 ) ( x k +1 , x k ) ( x k , x k − 1 ) �� �� f ( x 1 )+ f ( x 2 )+ f ( x 3 )+ f ( x 4 ) f · ω := ω 4 ( x 1 , x 2 , x 3 , x 4 ) ( x 1 , x 2 , x 3 , x 4 ) for ( x 1 , x 2 , x 3 , x 4 ) ∈ ♦ 2 .

Discrete Riemann Surfaces Linear theory External Product A form on ♦ can be averaged into a form on Λ:   � � � � � A ( α ♦ ) := 1 + + +  α ♦ , (4)   2  ( x , x ′ ) ( x , y ) ( y , x ′ ) ( x , y ′ ) ( y ′ , x ′ ) d �� �� A ( ω ♦ ) := 1 � ω ♦ , (5) 2 k =1 x ∗ ( x k , y k , x , y k − 1 ) Ker ( A ) = Span( d ♦ ε )

Discrete Riemann Surfaces Main Results Discrete Harmonicity ∗ : C k → C 2 − k , ◮ Hodge Star: � � ( y , y ′ ) ∗ α := ρ ( x , x ′ ) ( x , x ′ ) α . ◮ Discrete Laplacian: d ∗ := − ∗ d ∗ , ∆ := d d ∗ + d ∗ d , ∆ f ( x ) = � ρ ( x , x k ) ( f ( x ) − f ( x k )) . ◮ Hodge Decomposition: C k = Im d ⊕ ⊥ Im d ∗ ⊕ ⊥ 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