Generic Surface Model - Triangular Mesh Surfaces are represented as polyhedron triangular meshes. Isometric gluing of triangles in E 2 . Isometric gluing of triangles in H 2 , S 2 . David Gu Discrete Surface Ricci Flow
Generic Surface Model - Triangular Mesh Surfaces are represented as polyhedron triangular meshes. Isometric gluing of triangles in E 2 . Isometric gluing of triangles in H 2 , S 2 . David Gu Discrete Surface Ricci Flow
Generic Surface Model - Triangular Mesh Surfaces are represented as polyhedron triangular meshes. Isometric gluing of triangles in E 2 . Isometric gluing of triangles in H 2 , S 2 . David Gu Discrete Surface Ricci Flow
Discrete Generalization Concepts Discrete Riemannian Metric 1 Discrete Curvature 2 Discrete Conformal Metric Deformation 3 David Gu Discrete Surface Ricci Flow
Discrete Metrics Definition (Discrete Metric) A Discrete Metric on a triangular mesh is a function defined on the vertices, l : E = { all edges } → R + , satisfies triangular inequality. A mesh has infinite metrics. David Gu Discrete Surface Ricci Flow
Discrete Curvature Definition (Discrete Curvature) Discrete curvature: K : V = { vertices } → R 1 . K ( v ) = 2 π − ∑ α i , v �∈ ∂ M ; K ( v ) = π − ∑ α i , v ∈ ∂ M i i Theorem (Discrete Gauss-Bonnet theorem) K ( v )+ ∑ ∑ K ( v ) = 2 πχ ( M ) . v �∈ ∂ M v ∈ ∂ M v α 1 v α 2 α 3 α 1 α 2 David Gu Discrete Surface Ricci Flow
Discrete Metrics Determines the Curvatures v k v k v k θ k l j l j θ k l i l i l j θ k l i θ j θ i θ j θ i θ i θ j v i v j l k v j v i l k v i v j l k H 2 S 2 R 2 cosine laws cos θ i + cos θ j cos θ k cos l i = (1) sin θ j sin θ k cosh θ i + cosh θ j cosh θ k cosh l i = (2) sinh θ j sinh θ k cos θ i + cos θ j cos θ k 1 = (3) sin θ j sin θ k David Gu Discrete Surface Ricci Flow
Derivative cosine law Lemma (Derivative Cosine Law) Suppose corner angles are the 푣 푘 functions of edge lengths, then ∂θ i 휃 푘 l i 푙 푖 푙 푗 = ∂ l i A ∂θ i − ∂θ i cos θ k 휃 푖 = 휃 푗 ∂ l j ∂ l i 푣 푗 푣 푖 푙 푘 where A = l j l k sin θ i . David Gu Discrete Surface Ricci Flow
Discrete Conformal Structure David Gu Discrete Surface Ricci Flow
Discrete Conformal Metric Deformation Conformal maps Properties transform infinitesimal circles to infinitesimal circles. preserve the intersection angles among circles. Idea - Approximate conformal metric deformation Replace infinitesimal circles by circles with finite radii. David Gu Discrete Surface Ricci Flow
Discrete Conformal Metric Deformation vs CP David Gu Discrete Surface Ricci Flow
Discrete Conformal Metric Deformation vs CP David Gu Discrete Surface Ricci Flow
Discrete Conformal Metric Deformation vs CP David Gu Discrete Surface Ricci Flow
Thurston’s Circle Packing Metric φ ki γ i v i φ ij Thurston’s CP Metric l ki l ij We associate each vertex v i l jk v j v k with a circle with radius γ i . On γ j γ k edge e ij , the two circles φ jk intersect at the angle of Φ ij . The edge lengths are ij = γ 2 i + γ 2 j + 2 γ i γ j cos Φ ij l 2 CP Metric ( T , Γ , Φ) , T triangulation, Γ = { γ i |∀ v i } , Φ = { φ ij |∀ e ij } David Gu Discrete Surface Ricci Flow
Discrete Conformal Equivalence Metrics Definition Conformal Equivalence Two CP metrics ( T 1 , Γ 1 , Φ 1 ) and ( T 2 , Γ 2 , Φ 2 ) are conformal equivalent, if they satisfy the following conditions T 1 = T 2 and Φ 1 = Φ 2 . David Gu Discrete Surface Ricci Flow
Power Circle Definition (Power Circle) The unit circle orthogonal to three circles at the vertices 푣 푘 휏 푗푘 ( v i , γ i ) , ( v j , γ j ) and ( v k , γ k ) is 휏 푘푖 ℎ 푖 휃 푘 called the power circle. The 푙 푖 푙 푗 ℎ 푗 표 center is called the power ℎ 푘 휃 푗 휃 푖 center. The distance from the 푣 푗 푙 푘 푣 푖 power center to three edges are denoted as h i , h j , h k respectively. 휏 푖푗 David Gu Discrete Surface Ricci Flow
Derivative cosine law Theorem (Symmetry) d θ j d θ i = h k = du j du i l k d θ j d θ k = h i 푣 푘 휏 푗푘 = 휏 푘푖 du k du j l i 휃 푘 ℎ 푖 푙 푗 푙 푖 d θ k d θ i = h j ℎ 푗 표 = du i du k l j ℎ 푘 휃 푗 휃 푖 푣 푗 푙 푘 푣 푖 Therefore the differential 1-form ω = θ i du i + θ j du j + θ k du k is 휏 푖푗 closed. David Gu Discrete Surface Ricci Flow
Discrete Ricci Energy Definition (Discrete Ricci Energy) The functional associated with a CP metric on a triangle is � ( u i , u j , u k ) θ i ( u ) du i + θ j ( u ) du j + θ k ( u ) du k . E ( u ) = ( 0 , 0 , 0 ) Geometrical interpretation: the volume of a truncated hyperbolic hyper-ideal tetrahedron. 푣 푘 휏 푗푘 휏 푘푖 휃 푘 ℎ 푖 푙 푗 푙 푖 ℎ 푗 표 ℎ 푘 휃 푗 휃 푖 푣 푗 푣 푖 푙 푘 휏 푖푗 David Gu Discrete Surface Ricci Flow
Generalized Circle Packing/Pattern Definition (Tangential Circle Packing) l 2 ij = γ 2 i + γ 2 j + 2 γ i γ j . C k v k r k d ki d kj h i w j w i o h j d ik d jk h k r i r j C 0 C i C j v i w k v j d ij d ji David Gu Discrete Surface Ricci Flow
Generalized Circle Packing/Pattern Definition (Inversive Distance Circle Packing) ij = γ 2 i + γ 2 j + 2 γ i γ j η ij . l 2 where η ij > 1. C k v k θ k d ki τ jk C 0 τ ik d kj w j w i h j d jk τ ik o h i d ik τ jk h k d ij d ji v i v j τ ij w k τ ij C j θ i θ j C i l ij David Gu Discrete Surface Ricci Flow
Generalized Circle Packing/Pattern Definition (Discrete Yamabe Flow) l 2 ij = 2 γ i γ j η ij . where η ij > 0. v k τ ik τ kj d kj d ki θ k C 0 w j w i h j h i d jk d ik o τ ik τ kj h k θ i θ j v i v j w k d ji d ij τ ij τ ij David Gu Discrete Surface Ricci Flow
Voronoi Diagram Definition (Voronoi Diagram) Given p 1 , ··· , p k in R n , the Voronoi cell W i at p i is W i = { x || x − p i | 2 ≤ | x − p j | 2 , ∀ j } . The dual triangulation to the Voronoi diagram is called the Delaunay triangulation. David Gu Discrete Surface Ricci Flow
Power Distance Power Distance Given p i associated with 푝표푤 ( 푝 푖 , 푞 ) a sphere ( p i , r i ) the 푟 푖 power distance from 푞 q ∈ R n to p i is 푝 푖 pow ( p i , q ) = | p i − q | 2 − r 2 i . David Gu Discrete Surface Ricci Flow
Power Diagram Definition (Power Diagram) Given p 1 , ··· , p k in R n and sphere radii γ 1 , ··· , γ k , the power Voronoi cell W i at p i is W i = { x | Pow ( x , p i ) ≤ Pow ( x , p j ) , ∀ j } . The dual triangulation to Power diagram is called the Power Delaunay triangulation. David Gu Discrete Surface Ricci Flow
Voronoi Diagram Delaunay Triangulation Definition (Voronoi Diagram) Let ( S , V ) be a punctured surface, V is the vertex set. d is a flat cone metric, where the cone singularities are at the vertices. The Voronoi diagram is a cell decomposition of the surface, Voronoi cell W i at v i is W i = { p ∈ S | d ( p , v i ) ≤ d ( p , v j ) , ∀ j } . The dual triangulation to the voronoi diagram is called the Delaunay triangulation. David Gu Discrete Surface Ricci Flow
Power Voronoi Diagram Delaunay Triangulation Definition (Power Diagram) Let ( S , V ) be a punctured surface, with a generalized circle packing metric. The Power diagram is a cell decomposition of the surface, a Power cell W i at v i is W i = { p ∈ S | Pow ( p , v i ) ≤ Pow ( p , v j ) , ∀ j } . The dual triangulation to the power diagram is called the power Delaunay triangulation. David Gu Discrete Surface Ricci Flow
Edge Weight Definition (Edge Weight) ( S , V , d ) , d a generalized CP metric. D the Power diagram, T the Power Delaunay triangulation. ∀ e ∈ D , the dual edge ¯ e ∈ T , the weight w ( e ) = | e | e | . | ¯ David Gu Discrete Surface Ricci Flow
Discrete Surface Ricci Flow David Gu Discrete Surface Ricci Flow
Discrete Conformal Factor Conformal Factor Defined on each vertex u : V → R , log γ i R 2 logtanh γ i H 2 u i = 2 logtan γ i S 2 2 David Gu Discrete Surface Ricci Flow
Discrete Surface Ricci Flow Definition (Discrete Surface Ricci Flow with Surgery) Suppose ( S , V , d ) is a triangle mesh with a generalized CP metric, the discrete surface Ricci flow is given by du i dt = ¯ K i − K i , where ¯ K i is the target curvature. Furthermore, during the flow, the Triangulation preserves to be Power Delaunay. Theorem (Exponential Convergence) The flow converges to the target curvature K i ( ∞ ) = ¯ K i . Furthermore, there exists c 1 , c 2 > 0 , such that | K i ( t ) − K i ( ∞ ) | < c 1 e − c 2 t , | u i ( t ) − u i ( ∞ ) | < c 1 e − c 2 t , David Gu Discrete Surface Ricci Flow
Discrete Conformal Metric Deformation Properties Symmetry ∂ K i = ∂ K j = − w ij ∂ u j ∂ u i Discrete Laplace Equation dK i = ∑ w ij ( du i − du j ) [ v i , v j ] ∈ E namely d K = ∆ d u , David Gu Discrete Surface Ricci Flow
Discrete Laplace-Beltrami operator Definition (Laplace-Beltrami operator) ∆ is the discrete Lapalce-Beltrami operator, ∆ = ( d ij ) , where ∑ k w ik i = j d ij = − w ij i � = j , [ v i , v j ] ∈ E otherwise 0 Lemma Given ( S , V , d ) with generalized CP metric, if T is the Power Delaunay triangulation, then ∆ is positive definite on the linear space ∑ i u i = 0 . Because ∆ is diagonal dominant. David Gu Discrete Surface Ricci Flow
Discrete Surface Ricci Energy Definition (Discrete Surface Ricci Energy) Suppose ( S , V , d ) is a triangle mesh with a generalized CP metric, the discrete surface energy is defined as � u k ∑ ( ¯ E ( u ) = K i − K i ) du i . 0 i = 1 gradient ∇ E = ¯ K − K , 1 Hessian � ∂ 2 E 2 � = ∆ , ∂ u i ∂ u j Ricci flow is the gradient flow of the Ricci energy, 3 Ricci energy is concave, the solution is the unique global 4 maximal point, which can be obtained by Newton’s method. David Gu Discrete Surface Ricci Flow
Algorithm Input: a closed triangle mesh M , target curvature ¯ K , step length δ , threshold ε Output:a PL metric conformal to the original metric, realizing ¯ K . Initialize u i = 0, ∀ v i ∈ V . 1 compute edge length, corner angle, discrete curvature K i 2 update to Delaunay triangulation by edge swap 3 compute edge weight w ij . 4 u + = δ ∆ − 1 ( ¯ K − K ) 5 normalize u such that the mean of u i ’s is 0. 6 repeat step 2 through 6, until the max | ¯ K i − K i | < ε . 7 David Gu Discrete Surface Ricci Flow
Genus One Example David Gu Discrete Surface Ricci Flow
Hyperbolic Discrete Surface Yamabe Flow Discrete conformal metric deformation: u 1 θ 1 l 3 y 2 l 2 y 3 θ 3 u 2 θ 2 l 1 y 1 u 3 conformal factor y k e u i l k 2 e u j R 2 = 2 sinh y k e u i sinh l k 2 e u j H 2 = 2 sin y k e u i sin l k 2 e u j S 2 = 2 ∂ u j = ∂ K j Properties: ∂ K i ∂ u i and d K = ∆ d u . David Gu Discrete Surface Ricci Flow
Hyperbolic Discrete Surface Yamabe Flow Unified framework for both Discrete Ricci flow and Yamabe flow Curvature flow du dt = ¯ K − K , Energy � ∑ ( ¯ E ( u ) = K i − K i ) du i , i Hessian of E denoted as ∆ , d K = ∆ d u . David Gu Discrete Surface Ricci Flow
Genus Two Example David Gu Discrete Surface Ricci Flow
Genus Three Example David Gu Discrete Surface Ricci Flow
Existence Theorem David Gu Discrete Surface Ricci Flow
Delaunay Triangulation Definition (Delaunay Triangulation) Each PL metric d on ( S , V ) has a Delaunay triangulation T , such that for each edge e of T , a + a ′ ≤ π , a a ′ e It is the dual of Voronoi decomposition of ( S , V , d ) R ( v i ) = { x | d ( x , v j ) ≤ d ( x , v j ) for all v j } David Gu Discrete Surface Ricci Flow
Discrete Conformality Definition (Conformal change) Conformal factor u : V → R . Discrete conformal change is vertex scaling. u 1 vertex scaling l 2 e u 1 l 3 e u 2 e u 3 l 2 e u 1 l 3 u 3 u 2 l 1 e u 2 l 1 e u 3 proposed by physicists Rocek and Williams in 1984 in the Lorenzian setting. Luo discovered a variational principle associated to it in 2004. David Gu Discrete Surface Ricci Flow
Discrete Yamabe Flow Definition (Discrete Yamabe Flow) The discrete conformal factor deforms proportional to the difference between the target curvature and the current curvature du ( v i ) = ¯ K ( v i ) − K ( v i ) , dt the triangulation is updated to be Delaunay during the flow. David Gu Discrete Surface Ricci Flow
Discrete Conformality Definition (Discrete Conformal Equivalence) PL metrics d , d ′ on ( S , V ) are discrete conformal, d ∼ d ′ if there is a sequence d = d 1 , d 2 , ··· , d k = d ′ and T 1 , T 2 , ··· , T k on ( S , V ) , such that T i is Delaunay in d i 1 if T i � = T i + 1 , then ( S , d i ) ∼ = ( S , d i + 1 ) by an isometry 2 homotopic to id if T i = T i + 1 , ∃ u : V → R , such that ∀ edge e = [ v i , v j ] , 3 l d i + 1 ( e ) = e u ( v i ) l d i e u ( v j ) David Gu Discrete Surface Ricci Flow
Discrete Conformality Discrete conformal metrics v v mkc kc kc ka c a w kb mx w x b my y vertex scale diagonal switch vertex scale David Gu Discrete Surface Ricci Flow
Main Theorem Theorem (Gu-Luo-Sun-Wu (2013)) ∀ PL metrics d on closed ( S , V ) and ∀ ¯ K : V → ( − ∞ , 2 π ) , such that ∑ ¯ K ( v ) = 2 πχ ( S ) , ∃ a PL metric ¯ d, unique up to scaling on ( S , V ) , such that ¯ d is discrete conformal to d 1 The discrete curvature of ¯ d is ¯ K. 2 Furthermore, ¯ d can be found from d from a discrete curvature flow. Remark K = 2 πχ ( S ) ¯ , discrete uniformization. | V | David Gu Discrete Surface Ricci Flow
Main Theorem X. Gu, F. Luo, J. Sun, T. Wu, ”A discrete uniformization theorem The uniqueness of the solution is 1 for polyhedral surfaces”, obtained by the convexity of Journal of Differential discrete surface Ricci energy and Geometry, Volume 109, the convexity of the admissible Number 2 (2018), conformal factor space ( u -space). 223-256. The existence is given by the 2 (arXiv:1309.4175). equivalence between PL metrics on ( S , V ) and the decorated hyperbolic metrics on ( S , V ) and the Ptolemy identity. David Gu Discrete Surface Ricci Flow
PL Metric Teichm¨ uller Space David Gu Discrete Surface Ricci Flow
PL Metric Teichm¨ uller Space Definition (Marked Surface) Suppose Σ is a closed topological surface, V = { v 1 , v 2 ,..., v n } ⊂ Σ is a set of disjoint points on Σ , satisfying χ (Σ − V ) < 0. Definition (Metric Equivalence) Two polyhedral metrics d and d ′ are equivalent, if there is an isometric transformation h : (Σ , V , d ) → (Σ , V , d ′ ) , h is homotopic to the identity of the marked surface (Σ , V ) . Definition (PL Teichm¨ uller Space) All the equivalence classes of the PL metrics on the marked surface (Σ , V ) consist the Teichm¨ uller space T PL (Σ , V ) := { d | polyhedralmetricon (Σ , V ) } / { isometry ∼ id (Σ , V ) } . David Gu Discrete Surface Ricci Flow
PL Teichm¨ uller Space Definition (Local Chart for PL Teichm¨ uller Space) Assume T is a triangulation of (Σ , V ) , the edge length function determines a unique PL metric, Φ T : R E ( T ) → T PL (Σ , V ) , △ this gives a local coordinates of the PL Teichm¨ uller space, where the domain � � R E ( T ) x ∈ R E ( T ) |∀ ∆ = { e i , e j , e k } , x ( e i )+ x ( e j ) > x ( e k ) = △ > 0 is a convex set. We use P T to represent the image of Φ T , then ( P T , Φ − 1 T ) form a local chart of T PL (Σ , V ) . David Gu Discrete Surface Ricci Flow
PL Teichm¨ uller Space Definition (Atlas of PL metric Teichm¨ uller Space) Given a closed marked surface (Σ , V ) ,the atlas of T PL (Σ , V ) consists of all local charts ( P T , Φ − 1 T ) , where T exhaust all possible triangulations, ( P T , Φ − 1 � A ( T pl ( S , V )) = T ) . T From | V | + | F |−| E | = 2 − 2 g and 3 | F | = 2 | E | , we obtain | E | = 6 g − 6 + 3 | V | . Theorem (Troyanov) Given a closed marked surface (Σ , V ) , the PL metric Teichm¨ uller space T PL (Σ , V ) and the Euclidean space R 6 g − 6 + 3 | V | is diffeomorphic. David Gu Discrete Surface Ricci Flow
Complete Hyperbolic Metric Teichm¨ uller Space David Gu Discrete Surface Ricci Flow
Poincare Disk Model The unit disk is with hyperbolic Riemannian metric 4 | dz | 2 ds 2 = ( 1 −| z | 2 ) 2 , Figure: Hyperbolic geodesics in the Poincare model. David Gu Discrete Surface Ricci Flow
Upper Half Plane Model The upper half plane is with hyperbolic Riemannian metric ds 2 = dx 2 + dy 2 , y 2 ∞ f a b c 0 1 Figure: All hyperbolic ideal triangles are isometric¡ $ David Gu Discrete Surface Ricci Flow
Hyperbolic Ideal Quadrilateral Definition (Thurston’s Shear Coordinates) Given an ideal quadrilateral, Thurston’s shear coordinates equal to the oriented distance from L to R along the diagonal. ∞ R δ L L R ˜ B A ˜ R L − 1 0 t Figure: Hyperbolic Ideal Quadrilateral. David Gu Discrete Surface Ricci Flow
Hyperbolic Ideal Quadrilateral Definition (Thurston’s Shear Coordinates) Given an ideal quadrilateral, Thurston’s shear coordinates equal to the oriented distance from L to R along the diagonal. ∞ R δ L L R ˜ B A ˜ R L − 1 0 t Figure: Hyperbolic Ideal Quadrilateral. David Gu Discrete Surface Ricci Flow
Construction of Hyperbolic Metric Assume a genus g surface with n vertices removed, Σ = Σ g −{ v 1 , v 2 ,..., v n } , n ≥ 1, χ (Σ) < 0, (Σ , T ) is a triangulation. Given a function defined on edges, x : E ( T ) → R , construct a hyperbolic structure π ( X ) for every triangle ∆ ∈ T , construct a hyperbolic ideal 1 triangle, ∆ → ∆ ∗ ; for every edge e ∈ E ( T ) , adjacent to two faces 2 ∆ 1 ∩ ∆ 2 = e , glue two ideal triangles ∆ ∗ o ´ I ∆ ∗ 2 along e 1 isometrically, the shear coordinates on e equals to x ( e ) . x ( e ) e ∆ 1 ∆ 2 ∆ ∗ 1 ∆ ∗ 1 ∆ ∗ ∆ ∗ 2 2 Figure: Construction of a complete metric. David Gu Discrete Surface Ricci Flow
Ideal Triangulation Lemma If π ( x ) is a complete metric with finite area, namely each vertex becomes a cusp, then for each v ∈ { v 1 , v 2 ,..., v n } , ∑ x ( e ) = 0 . e ∼ v e 1 e 2 e 3 e 1 v z ′ z e 3 x 1 e 1 x 2 e 2 x 3 0 1 Figure: Condition for complete hyperbolic metric. David Gu Discrete Surface Ricci Flow
Hyperbolic Structure Define linear space: � x ∈ R E |∀ v ∈ V , ∑ � R E P = x ( e ) = 0 v ∼ e Theorem (Thurston) The mapping Φ T : R E P → T (Σ) , x �→ [ π ( x )] is injective and surjective, Φ T ( x ) under T has shear coordinates x ( e ) . v 2 v 1 (Σ , T ) David Gu Discrete Surface Ricci Flow v 1
Hyperbolic Teichm¨ uller Space Definition (Complete Hyperbolic Metric Teichm¨ uller Space) Given a closed marked surface (Σ , V ) with genus g , χ (Σ − V ) < 0, all the complete hyperbolic metrics defined on Σ − V with finite area, and each v ∈ V being a cusp, form the hyperbolic metric Teichm¨ uller space of Σ − V , denoted as T H (Σ , V ) . From | V | + | F |−| E | = 2 − 2 g and 3 | F | = 2 | E | , we obtain | E | = 6 g − 6 + 3 | V | . The cusp condition removes | V | freedoms. Corollary The hyperbolic metric Teichm¨ uller SpaceT (Σ , V ) is a real analytic manifold, diffeomorphic to R 6 g − 6 + 2 | V | , where g is the genus of the closed surface Σ . David Gu Discrete Surface Ricci Flow
Complete Hyperbolic Teichm¨ uller Space Definition (Complete Hyperbolic Metric Equivalence) Two complete hyperbolic metrics h and h ′ on a closed marked surface (Σ , V ) with finite total area are equivalent, if there is an isometric transformation h : (Σ − V , h ) → (Σ − V , d ′ ) , furthermore h is homotopic to the identity map of Σ − V . Definition (Complete Hyperbolic Teichm¨ uller Space) Given a closed marked surface (Σ , V ) , χ (Σ − V ) < 0, all the equivalence classes of the complete hyperbolic metrics with finite area on (Σ , V ) form the Teichm¨ uller space: T H (Σ − V ) = { h | h compelete , finitearea } / { isometry ∼ idof (Σ − V ) } (4) David Gu Discrete Surface Ricci Flow
Complete Hyperbolic Metric Teichm¨ uller Space Definition (Local Chart of T H (Σ − V ) ) Assume T is a triangulation of (Σ , V ) , its shear coordinates determines a unique complete hyperbolic metric with finite area, Θ T : Ω T → T H (Σ − V ) (5) this gives a local chart of the Teichm¨ uller space, where the domain Ω T is a sublinear space in R E ( T ) , satisfying the cusp conditions. Then (Ω T , Θ − 1 T ) form a local chart of T H (Σ − V ) . Definition (Atlas of T H (Σ − V ) ) Each triangulation T of (Σ , V ) corresponds to a local chart (Ω T , Θ − 1 T ) . By exhausting all possible triangulations, the union of all local charts gives the atlas of T H (Σ − V ) : � � Ω T , Θ − 1 � A ( T H (Σ − V )) = . T T David Gu Discrete Surface Ricci Flow
Decorated Hyperbolic Metric Teichm¨ uller Space David Gu Discrete Surface Ricci Flow
Decorated Ideal Hyperbolic Triangle τ is a decorated ideal hyperbolic triangle, three infinite vertices are v 1 , v 2 , v 3 ∈ ∂ H 2 . Each v i is associated with a horoball H i , the length of ∂ H i ∩ τ is α i ; the oriented length of e i is l i : if H j ∩ H k = / 0 then l i > 0, otherwise l i < 0. Penner’s λ -length L i is defined as 1 2 l i . L i := e α i α i l k l j l j l k α k α j α k α j l i l i Figure: Decorated ideal hyperbolic triangle, left frame l i > 0, right frame l i < 0. David Gu Discrete Surface Ricci Flow
Decorated Hyperbolic Metric Definition (Decorated Hyperbolic Metric) A decorated hyperbolic metric on a marked closed surface (Σ , V ) is represented as ( d , w ) : d is a complete, with finite area hyperbolic metric; 1 each cusp v i is associated with a haroball H i . The center of 2 H i is v i , the length of ∂ H i is w i . w = ( w 1 , w 2 ,..., w n ) ∈ R n > 0 H i ∂H i U i v i H j ∂H j U j v j Figure: Decorated hyperbolic metric. David Gu Discrete Surface Ricci Flow
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