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Voronoi Diagrams, Delaunay Triangulations and Polytopes Voronoi Diagrams, Delaunay Triangulations and Polytopes Jean-Daniel Boissonnat Geometrica, INRIA http://www-sop.inria.fr/geometrica Algorithmic Geometry Triangulations 2 Voronoi& Delaunay 1 / 1 Voronoi diagrams in nature Algorithmic Geometry Triangulations 2 Voronoi& Delaunay 2 / 1 The solar system (Descartes) Algorithmic Geometry Triangulations 2 Voronoi& Delaunay 3 / 1 Growth of merystem Algorithmic Geometry Triangulations 2 Voronoi& Delaunay 4 / 1 Euclidean Voronoi diagrams Voronoi cell V(pi) = x : x pi x pj ; j f k − k ≤ k − k 8 g Voronoi diagram (P)= collection of all cells V(pi), pi P f 2 g Algorithmic Geometry Triangulations 2 Voronoi& Delaunay 5 / 1 Voronoi diagrams and polytopes Polytope The intersection of a finite collection of half-spaces : = T h+ V i2I i Each Voronoi cell is a polytope The Voronoi diagram has the structure of a cell complex d+ The Voronoi diagram of P is the projection of a polytope of R 1 Algorithmic Geometry Triangulations 2 Voronoi& Delaunay 6 / 1 Voronoi diagrams and polytopes Polytope The intersection of a finite collection of half-spaces : = T h+ V i2I i Each Voronoi cell is a polytope The Voronoi diagram has the structure of a cell complex d+ The Voronoi diagram of P is the projection of a polytope of R 1 Algorithmic Geometry Triangulations 2 Voronoi& Delaunay 6 / 1 Voronoi diagrams and polytopes Polytope The intersection of a finite collection of half-spaces : = T h+ V i2I i Each Voronoi cell is a polytope The Voronoi diagram has the structure of a cell complex d+ The Voronoi diagram of P is the projection of a polytope of R 1 Algorithmic Geometry Triangulations 2 Voronoi& Delaunay 6 / 1 Voronoi diagrams and polytopes Polytope The intersection of a finite collection of half-spaces : = T h+ V i2I i Each Voronoi cell is a polytope The Voronoi diagram has the structure of a cell complex d+ The Voronoi diagram of P is the projection of a polytope of R 1 Algorithmic Geometry Triangulations 2 Voronoi& Delaunay 6 / 1 Voronoi diagrams and polyhedra Vor(p1;:::; pn) is the minimization diagram 2 of the n functions δi(x) = (x − pi) arg min(δi) = arg max(hi) 2 where hpi (x) = 2 pi · x − pi z = (x p )2 − i The minimization diagram of the δi is also pi the maximization diagram of the affine functions hpi (x) The faces of Vor(P) are the projections of T + the faces of V(P) = i hpi + 2 hpi = fx : xd+1 > 2pi · x − pi g Note ! the graph of hpi (x) is the hyperplane tangent 2 2 to : xd+1 = x at (x; x ) Q Algorithmic Geometry Triangulations 2 Voronoi& Delaunay 7 / 1 Voronoi diagrams and polyhedra Vor(p1;:::; pn) is the minimization diagram 2 of the n functions δi(x) = (x − pi) arg min(δi) = arg max(hi) 2 where hpi (x) = 2 pi · x − pi z = (x p )2 − i The minimization diagram of the δi is also pi the maximization diagram of the affine functions hpi (x) The faces of Vor(P) are the projections of T + the faces of V(P) = i hpi + 2 hpi = fx : xd+1 > 2pi · x − pi g Note ! the graph of hpi (x) is the hyperplane tangent 2 2 to : xd+1 = x at (x; x ) Q Algorithmic Geometry Triangulations 2 Voronoi& Delaunay 7 / 1 Voronoi diagrams and polyhedra Vor(p1;:::; pn) is the minimization diagram 2 of the n functions δi(x) = (x − pi) arg min(δi) = arg max(hi) 2 where hpi (x) = 2 pi · x − pi The minimization diagram of the δi is also the maximization diagram of the affine functions hpi (x) The faces of Vor(P) are the projections of T + the faces of V(P) = i hpi + 2 hpi = fx : xd+1 > 2pi · x − pi g Note ! the graph of hpi (x) is the hyperplane tangent 2 2 to : xd+1 = x at (x; x ) Q Algorithmic Geometry Triangulations 2 Voronoi& Delaunay 7 / 1 Voronoi diagrams and polyhedra Vor(p1;:::; pn) is the minimization diagram 2 of the n functions δi(x) = (x − pi) arg min(δi) = arg max(hi) 2 where hpi (x) = 2 pi · x − pi The minimization diagram of the δi is also the maximization diagram of the affine functions hpi (x) The faces of Vor(P) are the projections of T + the faces of V(P) = i hpi + 2 hpi = fx : xd+1 > 2pi · x − pi g Note ! the graph of hpi (x) is the hyperplane tangent 2 2 to : xd+1 = x at (x; x ) Q Algorithmic Geometry Triangulations 2 Voronoi& Delaunay 7 / 1 Voronoi diagrams and polyhedra Vor(p1;:::; pn) is the minimization diagram 2 of the n functions δi(x) = (x − pi) arg min(δi) = arg max(hi) 2 where hpi (x) = 2 pi · x − pi The minimization diagram of the δi is also the maximization diagram of the affine functions hpi (x) The faces of Vor(P) are the projections of T + the faces of V(P) = i hpi + 2 hpi = fx : xd+1 > 2pi · x − pi g Note ! the graph of hpi (x) is the hyperplane tangent 2 2 to : xd+1 = x at (x; x ) Q Algorithmic Geometry Triangulations 2 Voronoi& Delaunay 7 / 1 Voronoi diagrams and polytopes Lifting map The faces of Vor(P) are the projection of the faces of the polytope (P) = T h+ V i pi where hpi is the hyperplane tangent to paraboloid at the lifted 2 Q point (pi; pi ) Corollaries I The size of Vor( ) is the same as the size of (P) P V I Computing Vor( ) reduces to computing (P) P V Algorithmic Geometry Triangulations 2 Voronoi& Delaunay 8 / 1 Voronoi diagrams and polytopes Lifting map The faces of Vor(P) are the projection of the faces of the polytope (P) = T h+ V i pi where hpi is the hyperplane tangent to paraboloid at the lifted 2 Q point (pi; pi ) Corollaries I The size of Vor( ) is the same as the size of (P) P V I Computing Vor( ) reduces to computing (P) P V Algorithmic Geometry Triangulations 2 Voronoi& Delaunay 8 / 1 Voronoi diagram and Delaunay triangulation d Finite set of points P R 2 The Delaunay complex is the nerve of the Voronoi diagram d It is not always embedded in R Algorithmic Geometry Triangulations 2 Voronoi& Delaunay 9 / 1 Empty circumballs An (open) d-ball B circumscribing a simplex σ is called empty if ⊂ P 1 vert(σ) @B ⊂ 2 B = \P ; Del( ) is the collection of simplices admittingP an empty circumball Algorithmic Geometry Triangulations 2 Voronoi& Delaunay 10 / 1 Point sets in general position wrt spheres P = p1; p2 ::: pn is said to be in general position wrt spheres if d +f 2 points ofgP lying on a same (d 1)-sphere 6 9 − Theorem [Delaunay 1936] If P is in general position wrt spheres, the simplicial map f : vert(DelP) P ! provides a realization of Del(P) called the Delaunay triangulation of P. Algorithmic Geometry Triangulations 2 Voronoi& Delaunay 11 / 1 Point sets in general position wrt spheres P = p1; p2 ::: pn is said to be in general position wrt spheres if d +f 2 points ofgP lying on a same (d 1)-sphere 6 9 − Theorem [Delaunay 1936] If P is in general position wrt spheres, the simplicial map f : vert(DelP) P ! provides a realization of Del(P) called the Delaunay triangulation of P. Algorithmic Geometry Triangulations 2 Voronoi& Delaunay 11 / 1 Proof of Delaunay’s theorem 1 P Linearization h(σ) S(x) = x2 2c x + s; s = c2 r2 − · − z < 2c x s (h−) S(x) < 0 · − S , z = x2 ( ) P σ ^x = (x; x2) h− , 2 S Algorithmic Geometry Triangulations 2 Voronoi& Delaunay 12 / 1 Proof of Delaunay’s theorem 2 Proof of Delaunay’s th. P general position wrt spheres P^ in general position , σ a simplex, Sσ its circumscribing sphere σ Del(P) Sσ empty 2 , + i; p^i h , 8 2 Sσ σ^ is a face of conv−(P^) , Del(P) = proj(conv−(P^)) Algorithmic Geometry Triangulations 2 Voronoi& Delaunay 13 / 1 Proof of Delaunay’s theorem 2 Proof of Delaunay’s th. P general position wrt spheres P^ in general position , σ a simplex, Sσ its circumscribing sphere σ Del(P) Sσ empty 2 , + i; p^i h , 8 2 Sσ σ^ is a face of conv−(P^) , Del(P) = proj(conv−(P^)) Algorithmic Geometry Triangulations 2 Voronoi& Delaunay 13 / 1 Duality + − (P) = i h (P) = conv (P^) V \ pi D Algorithmic Geometry Triangulations 2 Voronoi& Delaunay 14 / 1 Voronoi diagrams, Delaunay triangulations and polytopes If P is in general position wrt spheres : + + duality − (P) = h ::: h (P) = conv ( p^1;:::; p^n ) V p1 \ \ pn −! D f g "# Voronoi Diagram of P nerve Delaunay Complex of P −! Algorithmic Geometry Triangulations 2 Voronoi& Delaunay 15 / 1 Combinatorial complexity The combinatorial complexity of the Voronoi diagram of n points of Rd is the same as the combinatorial complexity of the intersection of n half-spaces of Rd+1 The combinatorial complexity of the Delaunay triangulation of n points of Rd is the same as the combinatorial complexity of the convex hull of n points of Rd+1 The two complexities are the same by duality d 3 Θ(nd 2 e) Quadratic in R Algorithmic Geometry Triangulations 2 Voronoi& Delaunay 16 / 1 d Constructing Del(P), P = p1; :::; pn R f g ⊂ Algorithm 2 d+1 1 Lift the points of P onto the paraboloid xd+1 = x of R : 2 pi p^i = (pi; p ) ! i 2 Compute conv( p^i ) f g − d 3 Project the lower hull conv ( p^i ) onto R f g b d+1 c Complexity : Θ(n log n + n 2 ) Algorithmic Geometry Triangulations 2 Voronoi& Delaunay 17 / 1 Direct algorithm : insertion of a new point pi 1.
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