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

Conformal Equivalence of Triangle Meshes Boris Springborn with Peter Schröder und Ulrich Pinkall DFG Research Center M ATHEON Mathematics for key technologies Obergurgl, 17. Dezember 2008

introduction 1 smooth → discrete 2 some basics 3 discrete mapping problem 4 variational principle 5 piecewise projective interpolation 6 hyperbolic geometry 7 Conformal Equivalence of Triangle Meshes 2 / 32

Introduction 1 discrete version of conformal maps 2 polyhedral realization of hyperbolic cusp metrics Conformal Equivalence of Triangle Meshes 3 / 32

Introduction 1 discrete version of conformal maps 2 polyhedral realization of hyperbolic cusp metrics Conformal Equivalence of Triangle Meshes 3 / 32

Conformal maps conformal means angle preserving infinitesimal lengths scaled by conformal factor | df | = e u | dx | independent of direction in the small like similarity transformations Problem: conformally − − − − − − → plane surface in space Conformal Equivalence of Triangle Meshes 4 / 32

Example: Mercator’s projection (1569) Conformal Equivalence of Triangle Meshes 5 / 32

Example: Mercator’s projection (2008) Conformal Equivalence of Triangle Meshes 6 / 32

Example: Mercator’s projection (2008) Conformal Equivalence of Triangle Meshes 6 / 32

Example: Mercator’s projection (2008) Conformal Equivalence of Triangle Meshes 6 / 32

Example: Mercator’s projection (2008) Conformal Equivalence of Triangle Meshes 6 / 32

Example: Mercator’s projection (2008) Conformal Equivalence of Triangle Meshes 6 / 32

Example: Mercator’s projection (2008) Conformal Equivalence of Triangle Meshes 6 / 32

Example: Mercator’s projection (2008) Conformal Equivalence of Triangle Meshes 6 / 32

Example: Mercator’s projection (2008) Conformal Equivalence of Triangle Meshes 6 / 32

Example: Mercator’s projection (2008) Conformal Equivalence of Triangle Meshes 6 / 32

Example: Mercator’s projection (2008) Conformal Equivalence of Triangle Meshes 6 / 32

Example: Mercator’s projection (2008) Conformal Equivalence of Triangle Meshes 6 / 32

Example: Mercator’s projection (2008) Conformal Equivalence of Triangle Meshes 6 / 32

Example: Mercator’s projection (2008) Conformal Equivalence of Triangle Meshes 6 / 32

Example: Mercator’s projection (2008) Conformal Equivalence of Triangle Meshes 6 / 32

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

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

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

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

Examples Conformal Equivalence of Triangle Meshes 8 / 32

introduction 1 smooth → discrete 2 some basics 3 discrete mapping problem 4 variational principle 5 piecewise projective interpolation 6 hyperbolic geometry 7 Conformal Equivalence of Triangle Meshes 9 / 32

Smooth theory Definition Two Riemannian metrics g , ˜ g on a smooth manifold M are called conformally equivalent , if g = e 2 u g ˜ for some function u : M → R Gaussian curvatures e 2 u ˜ K = K + ∆ g u mapping problem ⇔ Given surface ( M , g ) , find conformally equivalent flat metric ˜ g Poisson problem ∆ g u = − K Conformal Equivalence of Triangle Meshes 10 / 32

Discrete abstract surface triangulation M = ( V , E , T ) Definition A discrete metric on M is a function ℓ : E → R > 0 , ij �→ ℓ ij satifying all triangle inequalities: ∀ ijk ∈ T : ℓ ij < ℓ jk + ℓ ki ℓ jk < ℓ ki + ℓ ij ℓ ki < ℓ ij + ℓ jk Conformal Equivalence of Triangle Meshes 11 / 32

Discrete Definition Two discrete metrics ℓ , ˜ ℓ on M are (discretely) conformally equivalent if 1 ˜ 2 ( u i + u j ) ℓ ij ℓ ij = e for some function u : V → R use λ ij = 2 log ℓ ij ℓ ij = e λ ij / 2 so ˜ and λ ij = λ ij + u i + u j Conformal Equivalence of Triangle Meshes 11 / 32

introduction 1 smooth → discrete 2 some basics 3 discrete mapping problem 4 variational principle 5 piecewise projective interpolation 6 hyperbolic geometry 7 Conformal Equivalence of Triangle Meshes 12 / 32

Two single triangles u 3 e λ 31 / 2 e λ 23 / 2 two single triangles are always conformally equivalent u 1 u 2 e λ 12 / 2 ˜ λ 12 = λ 12 + u 1 + u 2 ˜ λ 23 = λ 23 + u 2 + u 3 e ˜ λ 31 / 2 ˜ λ 31 = λ 31 + u 3 + u 1 e ˜ λ 23 / 2 e ˜ λ 12 / 2 Conformal Equivalence of Triangle Meshes 13 / 32

Two single triangles u 3 e λ 31 / 2 e λ 23 / 2 two single triangles are always conformally equivalent u 1 u 2 e λ 12 / 2 ˜ + λ 12 = λ 12 + u 1 + u 2 ˜ + λ 23 = λ 23 + u 2 + u 3 e ˜ λ 31 / 2 ˜ − λ 31 = λ 31 + u 3 + u 1 e ˜ λ 23 / 2 e ˜ λ 12 / 2 Conformal Equivalence of Triangle Meshes 13 / 32

Length cross ratio Definition For interior edges ij define length cross ratio lcr ij = ℓ ih ℓ jk ℓ hj ℓ ki ℓ , ˜ ℓ discretely conformally equivalent � � lcr ij = lcr ij Conformal Equivalence of Triangle Meshes 14 / 32

Teichmüller space ∀ interior vertices i : � lcr ij = 1 ij ∋ i discrete conformal strukture on M : equivalence class of discrete metrics M closed, compact, genus g : dim { conformal structures } = | E | − | V | = 6 g − 6 + 2 | V | = dim T g , | V | T g , n : Teichmüller space for genus g with n punctures Conformal Equivalence of Triangle Meshes 15 / 32

Möbius invariance immersion V → R n , i �→ v i induces discrete metric ℓ ij = � v i − v j � Möbius transformation: composition of inversions on spheres the only conformal transformations in R n if n ≥ 3 Möbius equivalent immersions induce conformally equivalent discrete metrics follows from � � � � � p q 1 1 � � � p − q � � p � 2 − � p � · � = � q � 2 � q � Conformal Equivalence of Triangle Meshes 16 / 32

introduction 1 smooth → discrete 2 some basics 3 discrete mapping problem 4 variational principle 5 piecewise projective interpolation 6 hyperbolic geometry 7 Conformal Equivalence of Triangle Meshes 17 / 32

Angles and curvatures lengths determine angles k jk = 2 tan − 1 � ( − ℓ ij + ℓ jk + ℓ ki )( ℓ ij + ℓ jk − ℓ ki ) α i ℓ ki ( ℓ ij − ℓ jk + ℓ ki )( ℓ ij + ℓ jk + ℓ ki ) ℓ jk α i angles sum around vertex i jk i j � ℓ ij α i Θ i = jk ijk ∋ i curvature at interior vertex i K i = 2 π − Θ i boundary curvature at boundary vertex κ i = π − Θ i Conformal Equivalence of Triangle Meshes 18 / 32

Mapping problem Discrete mapping problem 1 2 λ ij , and Given mesh M , metric ℓ ij = e desired angle sums � Θ i Find conformally equivalent metric ˜ ℓ ij with Θ i = � � Θ i � Θ i = 2 π for interior vertices (except for cone-like singulatrities) if ˜ ℓ ij are found, lay out triangles non-linear equations for u i Conformal Equivalence of Triangle Meshes 19 / 32

Mapping problem Discrete mapping problem 1 2 λ ij , and Given mesh M , metric ℓ ij = e desired angle sums � Θ i Find conformally equivalent metric ˜ ℓ ij with Θ i = � � Θ i � Θ i = 2 π for interior vertices (except for cone-like singulatrities) if ˜ ℓ ij are found, lay out triangles non-linear equations for u i looks bad Conformal Equivalence of Triangle Meshes 19 / 32

Mapping problem Discrete mapping problem 1 2 λ ij , and Given mesh M , metric ℓ ij = e desired angle sums � Θ i Find conformally equivalent metric ˜ ℓ ij with Θ i = � � Θ i � Θ i = 2 π for interior vertices (except for cone-like singulatrities) if ˜ ℓ ij are found, lay out triangles non-linear equations for u i looks bad variational principle comes to the rescue Conformal Equivalence of Triangle Meshes 19 / 32

introduction 1 smooth → discrete 2 some basics 3 discrete mapping problem 4 variational principle 5 piecewise projective interpolation 6 hyperbolic geometry 7 Conformal Equivalence of Triangle Meshes 20 / 32

Variational principle � � def ij ˜ jk ˜ ki ˜ α k α i α j S ( u ) = ˜ λ ij + ˜ λ jk + ˜ λ ki − π ( u i + u j + u k ) ijk ∈ T � � � α j α k α i + 2 L (˜ ij ) + 2 L (˜ jk ) + 2 L (˜ ki ) + Θ i u i i ∈ V Milnor’s Lobachevsky function � α L ( α ) = − log | 2 sin t | dt 0 ∂ S = � Θ i − � Θ i ∂ u i 2 ( λ ij + u i + u j ) solves mapping problem 1 ˜ ℓ ij = e � u = ( u 1 , . . . , u n ) is critical point of S ( u ) Conformal Equivalence of Triangle Meshes 21 / 32