Voronoi diagram on a Riemannian surface Aur elie Chapron ModalX - PowerPoint PPT Presentation

Case of the sphere Arbitrary surface Conclusion Voronoi diagram on a Riemannian surface Aur´ elie Chapron Modal’X (Paris Ouest) and LMRS (Rouen) 17 May 2016 Aur´ elie Chapron Modal’X (Paris Ouest) and LMRS (Rouen) Voronoi diagram on a Riemannian surface

Case of the sphere Arbitrary surface Conclusion Motivation Aim : Show a link between mean characteristics of the Voronoi cells and local characteristics of the surface image:R.Kunze Aur´ elie Chapron Modal’X (Paris Ouest) and LMRS (Rouen) Voronoi diagram on a Riemannian surface

Case of the sphere Arbitrary surface Conclusion Framework S Riemannian surface, with its Riemannian metric d , dx area measure induced by the metric, Φ Poisson point process of intensity λ dx and x 0 ∈ S added to Φ, The Voronoi cell of x 0 defined by C ( x 0 , Φ) = { y ∈ S , d ( x 0 , y ) ≤ d ( x , y ) , ∀ x ∈ Φ } N the number of vertices. Aur´ elie Chapron Modal’X (Paris Ouest) and LMRS (Rouen) Voronoi diagram on a Riemannian surface

Case of the sphere Arbitrary surface Conclusion Outline Case of the sphere 1 Arbitrary surface 2 Aur´ elie Chapron Modal’X (Paris Ouest) and LMRS (Rouen) Voronoi diagram on a Riemannian surface

Case of the sphere Arbitrary surface Conclusion Mean number of vertices wlog, assume x 0 to be the North pole on the sphere of constant 1 curvature K (of radius K ) √ E [ N ( C )] = 6 − 3 K � 3 K � πλ + e − 4 πλ πλ + 6 K Miles (1971) : n uniform points on the sphere Aur´ elie Chapron Modal’X (Paris Ouest) and LMRS (Rouen) Voronoi diagram on a Riemannian surface

Case of the sphere Arbitrary surface Conclusion Sketch of proof Step 1: characterize vertices of C Aur´ elie Chapron Modal’X (Paris Ouest) and LMRS (Rouen) Voronoi diagram on a Riemannian surface

Case of the sphere Arbitrary surface Conclusion Sketch of proof    � E [ N ( C )] = E 1 {B 1 ( x 0 , x 1 , x 2 ) ∩ Φ= ∅} + 1 {B 2 ( x 0 , x 1 , x 2 ) ∩ Φ= ∅}  x 1 , x 2 ∈ Φ Step 1: characterize vertices of C Aur´ elie Chapron Modal’X (Paris Ouest) and LMRS (Rouen) Voronoi diagram on a Riemannian surface

Case of the sphere Arbitrary surface Conclusion Sketch of proof E [ N ( C )] = λ 2 �� � e − λ vol( B 1 ( x 0 , x 1 , x 2 )) + e − λ vol( B 2 ( x 0 , x 1 , x 2 )) � dx 1 dx 2 2 x 1 , x 2 ∈S ( K ) Step 2: apply Mecke-Slivnyak formula Aur´ elie Chapron Modal’X (Paris Ouest) and LMRS (Rouen) Voronoi diagram on a Riemannian surface

Case of the sphere Arbitrary surface Conclusion Sketch of proof E [ N ( C )] = λ 2 � � e − λ vol( B 1 ( x 0 , x 1 , x 2 )) + e − λ vol( B 2 ( x 0 , x 1 , x 2 )) � 2 r 1 ,ϕ 1 , r 2 ,ϕ 2 √ √ × sin( Kr 1 ) sin( Kr 2 ) dr 1 d ϕ 1 dr 2 d ϕ 2 √ √ K K Step 3: use spherical coordinates Aur´ elie Chapron Modal’X (Paris Ouest) and LMRS (Rouen) Voronoi diagram on a Riemannian surface

Case of the sphere Arbitrary surface Conclusion Sketch of proof √ 2 � θ 1 � r 1 = √ arcsin(sin sin( KR )) 2 K √ 2 � θ 2 � r 2 = √ arcsin(sin sin( KR )) 2 K √ ϕ 1 = ϕ + π � θ 1 � 2 − arctan(tan cos( KR )) 2 √ ϕ 2 = ϕ + π � θ 2 � 2 − arctan(tan cos( KR )) 2 Step 4: make a Blaschke-Petkantschin type change of variables Aur´ elie Chapron Modal’X (Paris Ouest) and LMRS (Rouen) Voronoi diagram on a Riemannian surface

Case of the sphere Arbitrary surface Conclusion Sketch of proof √ KR )) � sin 3 ( π √ √ � KR ) KR )) + e − λ 2 π √ � e − λ 2 π 2 K E [ N ( C )] = 4 πλ 2 I K (1 − cos( K (1+cos( √ dR K 0 = 6 − 3 K � 6 + 3 K � πλ + e − 4 λπ K λπ where � θ 1 � � θ 2 � � � θ 1 − θ 2 �� � � � I = sin sin � sin � d θ 1 d θ 2 � � 2 2 2 θ 1 ,θ 2 ∈ [0 , 2 π ] Step 4: Make a Blaschke-Petkantschin type change of variables Aur´ elie Chapron Modal’X (Paris Ouest) and LMRS (Rouen) Voronoi diagram on a Riemannian surface

Case of the sphere Arbitrary surface Conclusion Strategy Find a way to adapt the method to a general surface Step 1: characterize vertices of C Step 2: apply Mecke-Slivnyak formula Step 3: use geodesic polar coordinates Step 4: make a Blaschke-Petkantschin type change of variables Step 5: find the volume of a geodesic ball image:R.Kunze Aur´ elie Chapron Modal’X (Paris Ouest) and LMRS (Rouen) Voronoi diagram on a Riemannian surface

Case of the sphere Arbitrary surface Conclusion Sketch of proof    � � E [ N ( C )] = E 1 {B ( x 0 , x 1 , x 2 ) ∩ Φ= ∅}  x 1 , x 2 ∈ Φ circumscribed balls Step 1: characterize vertices of C Aur´ elie Chapron Modal’X (Paris Ouest) and LMRS (Rouen) Voronoi diagram on a Riemannian surface

Case of the sphere Arbitrary surface Conclusion Sketch of proof E [ N ( C )] = λ 2 �� � e − λ vol( B ( x 0 , x 1 , x 2 )) dx 1 dx 2 2 x 1 , x 2 ∈ S circumscribed balls 1 Points ”far” from x 0 contribute negligibly. 2 For points around x 0 , we need similar changes of variables. Step 2: apply Mecke Slivnyak formula Aur´ elie Chapron Modal’X (Paris Ouest) and LMRS (Rouen) Voronoi diagram on a Riemannian surface

Case of the sphere Arbitrary surface Conclusion Exponential map Around x 0 , S can always be parametrized by its geodesic polar coordinates ( r , ϕ ), ie x = exp x 0 ( ru ϕ ) Step 3: use geodesic polar coordinates Aur´ elie Chapron Modal’X (Paris Ouest) and LMRS (Rouen) Voronoi diagram on a Riemannian surface

Case of the sphere Arbitrary surface Conclusion Rauch theorem dx = f ( r , ϕ ) drd ϕ Let K denote the Gaussian curvature. Rauch theorem (1951) Si 0 < δ ≤ K ≤ ∆ √ √ sin( ∆ r ) ≤ f ( r , ϕ ) ≤ sin( δ r ) √ √ ∆ δ Application: δ = K ( x 0 ) − ε , ∆ = K ( x 0 ) + ε Step 3: use geodesic polar coordinates Aur´ elie Chapron Modal’X (Paris Ouest) and LMRS (Rouen) Voronoi diagram on a Riemannian surface

Case of the sphere Arbitrary surface Conclusion Sketch of proof E [ N ( C )] = λ 2 � e − λ vol( B ( x 0 , x 1 , x 2 )) 2 ( r 1 ,ϕ 1 ) ( r 2 ,ϕ 2 ) r 1 − K ( x 0 ) r 3 r 2 − K ( x 0 ) r 3 dr 1 d ϕ 1 dr 2 d ϕ 2 + O ( e − c λ ) � � � � × + o ( r 3 + o ( r 3 1 1 ) 2 2 ) 6 6 Step 3: use geodesic polar coordinates Aur´ elie Chapron Modal’X (Paris Ouest) and LMRS (Rouen) Voronoi diagram on a Riemannian surface

Case of the sphere Arbitrary surface Conclusion Sketch of proof r 1 =? r 2 =? ϕ 1 =? ϕ 2 =? Step 4: make a Blaschke-Petkantschin type change of variables Aur´ elie Chapron Modal’X (Paris Ouest) and LMRS (Rouen) Voronoi diagram on a Riemannian surface

Case of the sphere Arbitrary surface Conclusion Toponogov theorem If δ ≤ K ≤ ∆ Step 4: make a Blaschke-Petkantschin type change of variables Aur´ elie Chapron Modal’X (Paris Ouest) and LMRS (Rouen) Voronoi diagram on a Riemannian surface

Case of the sphere Arbitrary surface Conclusion Sketch of proof r 1 = 2 sin( θ 1 / 2) R − K ( x 0 ) R 3 sin( θ 1 / 2) cos 2 ( θ 1 / 2) + o ( R 3 ) 3 r 2 = 2 sin( θ 2 / 2) R − K ( x 0 ) R 3 sin( θ 2 / 2) cos 2 ( θ 2 / 2) + o ( R 3 ) 3 2 + K ( x 0 ) R 2 ϕ 1 = ϕ + π 2 − θ 1 sin( θ 1 ) + o ( R 2 ) 4 2 + K ( x 0 ) R 2 ϕ 2 = ϕ + π 2 − θ 2 sin( θ 2 ) + o ( R 2 ) 4 Step 4: make a Blaschke-Petkantschin type change of variables Aur´ elie Chapron Modal’X (Paris Ouest) and LMRS (Rouen) Voronoi diagram on a Riemannian surface

Case of the sphere Arbitrary surface Conclusion Sketch of proof � � E [ N ( C )] = 2 λ 2 I e − λ vol( B ( z , R )) � R 3 − K ( x 0 ) R 5 � dRd ϕ + O ( e − c λ ) + o ( R 5 ) 2 R ϕ where � � �� � � θ 1 � � θ 2 � θ 1 − θ 2 � � I = sin sin � sin � d θ 1 d θ 2 � � 2 2 2 θ 1 ,θ 2 Step 4: make a Blaschke-Petkantschin type change of variables Aur´ elie Chapron Modal’X (Paris Ouest) and LMRS (Rouen) Voronoi diagram on a Riemannian surface

Case of the sphere Arbitrary surface Conclusion Volume of small geodesic balls Bertrand-Diquet-Puiseux theorem (1848) When r → 0, x ∈ S vol( B ( z , r )) = π r 2 − K ( z ) π r 4 + o ( r 4 ) 12 Step 5: find the volume of the circumscribed ball Aur´ elie Chapron Modal’X (Paris Ouest) and LMRS (Rouen) Voronoi diagram on a Riemannian surface

Case of the sphere Arbitrary surface Conclusion Result e − λ ( π R 2 − π K ( x 0) R 4 E [ N ( C )] = 12 π 2 λ 2 � R max + o ( R 4 )) × [ R 3 − K ( x 0 ) R 5 + o ( R 5 )] dR + O ( e − c λ ) 12 2 0 When λ goes to infinity, Laplace’s method yields Mean number of vertices � 1 � E [ N ( C )] = 6 − 3 K ( x 0 ) + o πλ λ Aur´ elie Chapron Modal’X (Paris Ouest) and LMRS (Rouen) Voronoi diagram on a Riemannian surface

Case of the sphere Arbitrary surface Conclusion Take Home Message On surfaces: ֒ → Link between mean number of vertices and Gaussian curvature ֒ → Result available for surface of negative curvature (Isokawa 2000) ֒ → Other mean characteristics: area, perimeter Ongoing work on dimension ≥ 3 : ֒ → Link between mean number of vertices and scalar curvature ֒ → Perspective: other characteristics to get other curvatures ֒ → . . . Aur´ elie Chapron Modal’X (Paris Ouest) and LMRS (Rouen) Voronoi diagram on a Riemannian surface