Voronoi Diagrams, Delaunay Triangulations and Polytopes Jean-Daniel - PowerPoint PPT Presentation

Voronoi Diagrams, Delaunay Triangulations and Polytopes Jean-Daniel Boissonnat Geometrica, INRIA http://www-sop.inria.fr/geometrica Winter School, University of Nice Sophia Antipolis January 26-30, 2015 Winter School 2 Voronoi, Delaunay & Polytopes Sophia Antipolis 1 / 43

Voronoi diagrams in nature Winter School 2 Voronoi, Delaunay & Polytopes Sophia Antipolis 2 / 43

The solar system (Descartes) Winter School 2 Voronoi, Delaunay & Polytopes Sophia Antipolis 3 / 43

Growth of merystem Winter School 2 Voronoi, Delaunay & Polytopes Sophia Antipolis 4 / 43

Euclidean Voronoi diagrams Voronoi cell V ( p i ) = { x : � x − p i � ≤ � x − p j � , ∀ j } Voronoi diagram ( P ) = { collection of all cells V ( p i ) , p i ∈ P } Winter School 2 Voronoi, Delaunay & Polytopes Sophia Antipolis 5 / 43

Voronoi diagrams and polytopes Polytope i ∈ I h + V = � The intersection of a finite collection of half-spaces : i Each Voronoi cell is a polytope The Voronoi diagram has the structure of a cell complex The Voronoi diagram of P is the projection of a polytope of R d + 1 Winter School 2 Voronoi, Delaunay & Polytopes Sophia Antipolis 6 / 43

Voronoi diagrams and polytopes Polytope i ∈ I h + V = � The intersection of a finite collection of half-spaces : i Each Voronoi cell is a polytope The Voronoi diagram has the structure of a cell complex The Voronoi diagram of P is the projection of a polytope of R d + 1 Winter School 2 Voronoi, Delaunay & Polytopes Sophia Antipolis 6 / 43

Voronoi diagrams and polytopes Polytope i ∈ I h + V = � The intersection of a finite collection of half-spaces : i Each Voronoi cell is a polytope The Voronoi diagram has the structure of a cell complex The Voronoi diagram of P is the projection of a polytope of R d + 1 Winter School 2 Voronoi, Delaunay & Polytopes Sophia Antipolis 6 / 43

Voronoi diagrams and polytopes Polytope i ∈ I h + V = � The intersection of a finite collection of half-spaces : i Each Voronoi cell is a polytope The Voronoi diagram has the structure of a cell complex The Voronoi diagram of P is the projection of a polytope of R d + 1 Winter School 2 Voronoi, Delaunay & Polytopes Sophia Antipolis 6 / 43

Voronoi diagrams and polyhedra Vor ( p 1 , . . . , p n ) is the minimization diagram of the n functions δ i ( x ) = ( x − p i ) 2 arg min ( δ i ) = arg max ( h i ) where h p i ( x ) = 2 p i · x − p 2 i z = ( x − p i ) 2 The minimization diagram of the δ i is also p i the maximization diagram of the affine functions h p i ( x ) The faces of Vor ( P ) are the projections of i h + the faces of V ( P ) = � p i h + p i = { x : x d + 1 > 2 p i · x − p 2 i } Note ! the graph of h p i ( x ) is the hyperplane tangent x d + 1 = x 2 ( x , x 2 ) to Q : at Winter School 2 Voronoi, Delaunay & Polytopes Sophia Antipolis 7 / 43

Voronoi diagrams and polyhedra Vor ( p 1 , . . . , p n ) is the minimization diagram of the n functions δ i ( x ) = ( x − p i ) 2 arg min ( δ i ) = arg max ( h i ) where h p i ( x ) = 2 p i · x − p 2 i z = ( x − p i ) 2 The minimization diagram of the δ i is also p i the maximization diagram of the affine functions h p i ( x ) The faces of Vor ( P ) are the projections of i h + the faces of V ( P ) = � p i h + p i = { x : x d + 1 > 2 p i · x − p 2 i } Note ! the graph of h p i ( x ) is the hyperplane tangent x d + 1 = x 2 ( x , x 2 ) to Q : at Winter School 2 Voronoi, Delaunay & Polytopes Sophia Antipolis 7 / 43

Voronoi diagrams and polyhedra Vor ( p 1 , . . . , p n ) is the minimization diagram of the n functions δ i ( x ) = ( x − p i ) 2 arg min ( δ i ) = arg max ( h i ) where h p i ( x ) = 2 p i · x − p 2 i The minimization diagram of the δ i is also the maximization diagram of the affine functions h p i ( x ) The faces of Vor ( P ) are the projections of i h + the faces of V ( P ) = � p i h + p i = { x : x d + 1 > 2 p i · x − p 2 i } Note ! the graph of h p i ( x ) is the hyperplane tangent x d + 1 = x 2 ( x , x 2 ) to Q : at Winter School 2 Voronoi, Delaunay & Polytopes Sophia Antipolis 7 / 43

Voronoi diagrams and polyhedra Vor ( p 1 , . . . , p n ) is the minimization diagram of the n functions δ i ( x ) = ( x − p i ) 2 arg min ( δ i ) = arg max ( h i ) where h p i ( x ) = 2 p i · x − p 2 i The minimization diagram of the δ i is also the maximization diagram of the affine functions h p i ( x ) The faces of Vor ( P ) are the projections of i h + the faces of V ( P ) = � p i h + p i = { x : x d + 1 > 2 p i · x − p 2 i } Note ! the graph of h p i ( x ) is the hyperplane tangent x d + 1 = x 2 ( x , x 2 ) to Q : at Winter School 2 Voronoi, Delaunay & Polytopes Sophia Antipolis 7 / 43

Voronoi diagrams and polyhedra Vor ( p 1 , . . . , p n ) is the minimization diagram of the n functions δ i ( x ) = ( x − p i ) 2 arg min ( δ i ) = arg max ( h i ) where h p i ( x ) = 2 p i · x − p 2 i The minimization diagram of the δ i is also the maximization diagram of the affine functions h p i ( x ) The faces of Vor ( P ) are the projections of i h + the faces of V ( P ) = � p i h + p i = { x : x d + 1 > 2 p i · x − p 2 i } Note ! the graph of h p i ( x ) is the hyperplane tangent x d + 1 = x 2 ( x , x 2 ) to Q : at Winter School 2 Voronoi, Delaunay & Polytopes Sophia Antipolis 7 / 43

Voronoi diagrams and polytopes Lifting map The faces of Vor ( P ) are the projection of the faces of the polytope i h + V ( P ) = � p i where h p i is the hyperplane tangent to paraboloid Q at the lifted point ( p i , p 2 i ) Corollaries ◮ The size of Vor ( P ) is the same as the size of V ( P ) ◮ Computing Vor ( P ) reduces to computing V ( P ) Winter School 2 Voronoi, Delaunay & Polytopes Sophia Antipolis 8 / 43

Voronoi diagrams and polytopes Lifting map The faces of Vor ( P ) are the projection of the faces of the polytope i h + V ( P ) = � p i where h p i is the hyperplane tangent to paraboloid Q at the lifted point ( p i , p 2 i ) Corollaries ◮ The size of Vor ( P ) is the same as the size of V ( P ) ◮ Computing Vor ( P ) reduces to computing V ( P ) Winter School 2 Voronoi, Delaunay & Polytopes Sophia Antipolis 8 / 43

Polytopes (convex polyhedra) Two ways of defining polytopes Convex hull of a finite set of points : V = conv ( P ) H = ∩ h ∈ H h + Intersection of a finite set of half-spaces : i Winter School 2 Voronoi, Delaunay & Polytopes Sophia Antipolis 9 / 43

Facial structure of a polytope Supporting hyperplane h : H ∩ P � = ∅ P on one side of h Faces : P ∩ h , h supp. hyp. Dimension of a face : the dim. of its affine hull Winter School 2 Voronoi, Delaunay & Polytopes Sophia Antipolis 10 / 43

General position Points in general position ◮ P is in general position iff no subset of k + 2 points lie in a k -flat ⇒ If P is in general position, all faces of conv ( P ) are simplices Hyperplanes in general position ◮ H is in general position iff the intersection of any subset of d − k hyperplanes intersect in a k -flat ⇒ any k -face is the intersection of d − k hyperplanes Winter School 2 Voronoi, Delaunay & Polytopes Sophia Antipolis 11 / 43

General position Points in general position ◮ P is in general position iff no subset of k + 2 points lie in a k -flat ⇒ If P is in general position, all faces of conv ( P ) are simplices Hyperplanes in general position ◮ H is in general position iff the intersection of any subset of d − k hyperplanes intersect in a k -flat ⇒ any k -face is the intersection of d − k hyperplanes Winter School 2 Voronoi, Delaunay & Polytopes Sophia Antipolis 11 / 43

Duality between points and hyperplanes h : x d = a · x ′ − b h ∗ = ( a , b ) ∈ R d hyperplane of R d − → point p ∗ ⊂ R d p = ( p ′ , p d ) ∈ R d point − → hyperplane = { ( a , b ) ∈ R d : b = p ′ · a − p d } Duality ◮ preserves incidences : p d = a · p ′ − b ⇐ ⇒ b = p ′ · a − p d ⇐ ⇒ h ∗ ∈ p ∗ p ∈ h ⇐ ⇒ p d > a · p ′ − b ⇐ ⇒ b > p ′ · a − p d ⇐ ⇒ h ∗ ∈ p ∗ + p ∈ h + ⇐ ⇒ ◮ is an involution and thus is bijective : h ∗∗ = h and p ∗∗ = p Winter School 2 Voronoi, Delaunay & Polytopes Sophia Antipolis 12 / 43

Duality between points and hyperplanes h : x d = a · x ′ − b h ∗ = ( a , b ) ∈ R d hyperplane of R d − → point p ∗ ⊂ R d p = ( p ′ , p d ) ∈ R d point − → hyperplane = { ( a , b ) ∈ R d : b = p ′ · a − p d } Duality ◮ preserves incidences : p d = a · p ′ − b ⇐ ⇒ b = p ′ · a − p d ⇐ ⇒ h ∗ ∈ p ∗ p ∈ h ⇐ ⇒ p d > a · p ′ − b ⇐ ⇒ b > p ′ · a − p d ⇐ ⇒ h ∗ ∈ p ∗ + p ∈ h + ⇐ ⇒ ◮ is an involution and thus is bijective : h ∗∗ = h and p ∗∗ = p Winter School 2 Voronoi, Delaunay & Polytopes Sophia Antipolis 12 / 43

Duality between polytopes Let h 1 , . . . , h n be n hyperplanes of R d and let H = ∩ h + i h 3 h ∗ 3 h 1 * h 2 h ∗ 1 s s h ∗ 2 A vertex s of H is ¯ the intersection of k ≥ d hyperplanes h 1 , . . . , h k lying above all the other hyperplanes ⇒ s ∗ is a hyperplane that 1 . contains h ∗ 1 , . . . , h ∗ = k H ∗ = conv − ( h ∗ 1 , . . . , h ∗ 2 . supports k ) General position s ∗ supports a ( d − 1 ) -simplex de H ∗ ⇒ s is the intersection of d hyperplanes Winter School 2 Voronoi, Delaunay & Polytopes Sophia Antipolis 13 / 43