Ioannis Emiris and Vissarion Fisikopoulos
Transcript of Ioannis Emiris and Vissarion Fisikopoulos
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Voronoi diagram and Delaunay triangulation
Ioannis Emirisand Vissarion Fisikopoulos
Dept. of Informatics & Telecommunications, University of Athens
Computational Geometry, Spring 2018
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Outline
1 Voronoi diagram
2 Delaunay triangulation
3 PropertiesEmpty circleComplexityMin max angle
4 Algorithms and complexityIncremental DelaunayFurther algorithms
5 (Generalizations and Representation)
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Outline
1 Voronoi diagram
2 Delaunay triangulation
3 PropertiesEmpty circleComplexityMin max angle
4 Algorithms and complexityIncremental DelaunayFurther algorithms
5 (Generalizations and Representation)
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Example and definition
Sites: P := p1, . . . , pn ⊂ R2
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Example and definitionSites: P := p1, . . . , pn ⊂ R2
Voronoi cell: q ∈ V (pi ) ⇔ dist(q, pi ) ≤ dist(q, pj), ∀pj ∈ P, j 6= i
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Example and definitionSites: P := p1, . . . , pn ⊂ R2
Voronoi cell: q ∈ V (pi ) ⇔ dist(q, pi ) ≤ dist(q, pj), ∀pj ∈ P, j 6= i
Georgy F. Voronoy(1868 - 1908)
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Faces of Voronoi diagram
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Faces of Voronoi diagram
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Faces of Voronoi diagram
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Faces of Voronoi diagram
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Faces of Voronoi diagram
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Voronoi diagram
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Formalization
• sites: points P = p1, . . . , pn ⊂ R2.
• Voronoi cell/region V (pi ) of site pi :
q ∈ V (pi ) ⇔ dist(q, pi ) ≤ dist(q, pj), ∀pj ∈ P, j 6= i .
• Voronoi edge is the common boundary of two adjacent cells.• Voronoi vertex is the common boundary of 3 adjacent cells, orthe intersection of ≥ 2 (hence ≥ 3) Voronoi edges.Generically, of exactly 3 Voronoi edges.
Voronoi diagram of P = dual of Delaunay triangulation of P.• Voronoi cell ↔ vertex of Delaunay triangles: site• neighboring cells (Voronoi edge) ↔ Delaunay edge, defined bycorresponding sites (line of Voronoi edge ⊥ line of Delaunay edge)• Voronoi vertex ↔ Delaunay triangle.
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Outline
1 Voronoi diagram
2 Delaunay triangulation
3 PropertiesEmpty circleComplexityMin max angle
4 Algorithms and complexityIncremental DelaunayFurther algorithms
5 (Generalizations and Representation)
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Triangulation
A triangulation of a pointset (sites) P ⊂ R2 is a collection oftriplets from P, namely triangles, s.t.
I The union of the triangles covers the convex hull of P.
I Every pair of triangles intersect at a (possibly empty) commonface (∅, vertex, edge).
I Usually (CGAL): Set of triangle vertices = P.
Example: P, incomplete, invalid, subdivision, triangulation.
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Delaunay Triangulation: dual of Voronoi diagram
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Delaunay Triangulation: dual of Voronoi diagram
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Delaunay Triangulation: dual of Voronoi diagram
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Delaunay Triangulation: dual of Voronoi diagram
Boris N. Delaunay(1890 - 1980)
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Delaunay triangulation: projection from parabola
Definition/Construction of Delaunay triangulation:
I Lift sites p = (x) ∈ R to p = (x , x2) ∈ R2
I Compute the convex hull of the lifted points
I Project the lower hull to R
y = x2
p1 p2 p3 p4
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Delaunay triangulation: going a bit higher. . .
Definition/Construction of Delaunay triangulation:
I Lift sites p = (x , y) ∈ R2 to p = (x , y , x2+y2) ∈ R3
I Compute the convex hull of the lifted points
I Project the lower hull to R2: arbitrarily triangulate lowerfacets that are polygons (not triangles)
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Nearest NeighborsReconstructionMeshing
Applications
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Voronoi by Lift & Project
Lifting:– Consider the paraboloid x3 = x21 + x22 .– For every site p, consider its lifted image p on the parabola.– Given p, ∃ unique (hyper)plane tangent to the parabola at p.
Project:– For every (hyper)plane, consider the halfspace above.– The intersection of halfspaces is a (unbounded) convex polytope– Its Lower Hull projects bijectively to the Voronoi diagram.
Proof:– Let E : x21 + x22 − x3 = 0 be the paraboloid equation.
– ∇E (a) =(∂E∂x1, ∂E∂x2 ,
∂E∂x3
)a= (2a1, 2a2,−1).
– Point x ∈ plane h(x) ⇔ (x − a) · ∇E (a) = 0 ⇔2a1(x1 − a1) + 2a2(x2 − a2) − (x3 − a3) = 0, which is h’s equation.
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Lift & Project in 1D
pp’ q
h
h’
y=0
y=x^2
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Outline
1 Voronoi diagram
2 Delaunay triangulation
3 PropertiesEmpty circleComplexityMin max angle
4 Algorithms and complexityIncremental DelaunayFurther algorithms
5 (Generalizations and Representation)
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Main Delaunay property: empty circle/sphere
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Main Delaunay property: empty circle/sphere
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Main Delaunay property: empty circle/sphere
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Main Delaunay property: 1 picture proof
Thm (in R): S(p1, p2) is a Delaunay segment ⇔ its interiorcontains no pi .
Proof. Delaunay segment ⇔ (p1, p2) edge of the Lower Hull⇔ no pi “below” (p1, p2) on the parabola⇔ no pi inside the segment (p1, p2).
y = x2
p1 p2 p3 p4
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Main Delaunay property: 1 picture proof
Thm (in R2): T (p1, p2, p3) is a Delaunay triangle ⇔ the interior ofthe circle through p1, p2, p3 (enclosing circle) contains no pi .
Proof. Circle(p1, p2, p3) contains no pi in interior⇔ plane of lifted p1, p2, p3 leaves all lifted pi on same halfspace⇔ CCW(p1, p2, p3, pi ) of same sign for all i .Suffices to prove: pi lies on Circle(p1, p2, p3)⇔ pi lies on plane of p1, p2, p3 ⇔ CCW(p1, p2, p3, pi ) = 0.
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Predicate InCircle
Given points p, q, r , s ∈ R2, point s = (sx , sy ) lies inside the circlethrough p, q, r ⇔
det
px py p2x + p2y 1qx qy q2x + q2y 1rx ry r2x + r2y 1sx sy s2x + s2y 1
> 0,
assuming p, q, r in clockwise order (otherwise det < 0).
Lemma. InCircle(p, q, r , s) = 0 ⇔ ∃ circle through p, q, r , s.Proof. InCircle(p, q, r , s) = 0 ⇔ CCW (p, q, r , s) = 0
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Delaunay faces
Theorem. Let P be a set of sites ∈ R2:
(i) Sites pi , pj , pk ∈ P are vertices of a Delaunay triangle ⇔ thecircle through pi , pj , pk contains no site of P in its interior.
(ii) Sites pi , pj ∈ P form an edge of the Delaunay triangulation ⇔there is a closed disc C that contains pi , pj on its boundaryand does not contain any other site of P.
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Outline
1 Voronoi diagram
2 Delaunay triangulation
3 PropertiesEmpty circleComplexityMin max angle
4 Algorithms and complexityIncremental DelaunayFurther algorithms
5 (Generalizations and Representation)
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Triangulations of planar pointsets
Thm. Let P be set of n points in R2, not all colinear, k = #pointson boundary of CH(P). Any triangulation of P has 2n − 2 − ktriangles and 3n − 3 − k edges.
Proof.
I f: #facets (except ∞)
I e: #edges
I n: #vertices
1. Euler: n − e + (f + 1) − 1 = 1; for d-polytope:∑di=0(−1)i fi = 1
2. Any planar triangulation: total degree = 3f + k = 2e.
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Properties of Voronoi diagram
Lemma. |V | ≤ 2n − 5, |E | ≤ 3n − 6, n = |P |,by Euler’s theorem for planar graphs: |V |− |E |+ n − 1 = 1.
Max Empty Circle CP(q) centered at q: no interior site pi ∈ P.Lem: q ∈ R2 is Voronoi vertex ⇔ C (q) has ≥ 3 sites on perimeter
Any perpendicular bisector of segment (pi , pj) defines a Voronoiedge ⇔ ∃ q on bisector s.t. C (q) has only pi , pj on perimeter
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Outline
1 Voronoi diagram
2 Delaunay triangulation
3 PropertiesEmpty circleComplexityMin max angle
4 Algorithms and complexityIncremental DelaunayFurther algorithms
5 (Generalizations and Representation)
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Delaunay maximizes the smallest angle
Let T be a triangulation with m triangles.Sort the 3m angles: a1 6 a2 6 · · · 6 a3m. Ta := a1, a2, . . . , a3m.Edge e = (pi , pj) is illegal ⇔ min16i66 ai < min16i66 a
′i .
pl
pk
pj
pi
pl
pk
pj
pi
a a′
T ′ obtained from T by flipping illegal e, then T ′a >lex Ta.
Flips yield triangulation without illegal edges.The algorithm terminates (angles decrease), but is O(n2).
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Insertion by flips
∆1
∆2
∆3
pr
∆2
∆3
pi
pj
∆5
∆4
∆3
pi pk
∆7
∆4
∆6
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Outline
1 Voronoi diagram
2 Delaunay triangulation
3 PropertiesEmpty circleComplexityMin max angle
4 Algorithms and complexityIncremental DelaunayFurther algorithms
5 (Generalizations and Representation)
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Lower bound
Ω(n log n) by reduction from sorting
(xi, x2i )
xi
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Delaunay triangulation
Theorem. Let P be a set of points ∈ R2. A triangulation T of Phas no illegal edge ⇔ T is a Delaunay triangulation of P.
Cor. Constructing the Delaunay triangulation is a fast (optimal)way of maximizing the min angle.
Algorithms in R2:– Lift, CH3, project the lower hull: O(n log n)– Incremental algorithm: O(n log n) exp., O(n2) worst– Voronoi diagram (Fortune’s sweep): O(n log n)– Divide + Conquer: O(n log n)
See Voronoi algo’s below.
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Outline
1 Voronoi diagram
2 Delaunay triangulation
3 PropertiesEmpty circleComplexityMin max angle
4 Algorithms and complexityIncremental DelaunayFurther algorithms
5 (Generalizations and Representation)
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Incremental Delaunay
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Incremental Delaunay
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Incremental Delaunay
Find triangles in conflict
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Incremental Delaunay
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Incremental Delaunay
Delete triangles in conflict
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Incremental Delaunay
Triangulate hole
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Outline
1 Voronoi diagram
2 Delaunay triangulation
3 PropertiesEmpty circleComplexityMin max angle
4 Algorithms and complexityIncremental DelaunayFurther algorithms
5 (Generalizations and Representation)
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Divide & Conquer
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Fortune’s sweep
2 events:
pi
pj
pk
C
L
v
pi
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Outline
1 Voronoi diagram
2 Delaunay triangulation
3 PropertiesEmpty circleComplexityMin max angle
4 Algorithms and complexityIncremental DelaunayFurther algorithms
5 (Generalizations and Representation)
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General dimension polytopes
Faces of a polytope are polytopes forming its extreme elements.A facet of a d-dimensional polytope is (d − 1)-dimensional face:• The facets of a segment are vertices (0-faces).• The facets of a polygon are edges (1-faces)• The facets of a 3-polyhedron are polygons.• The facets of a 4d polytope are 3d polytopes.
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General dimension triangulation
A triangulation of a pointset (sites) P ⊂ Rd is a collection of(d + 1)-tuples from P, namely simplices, s.t.
I The union of the simplices covers the convex hull of P.
I Every pair of simplices intersect at a (possibly empty)common face.
I Usually: Set of simplex vertices = P.
I Delaunay: no site lies in the circum-hypersphere inscribing anysimplex of the triangulation.
In 3d, two simplices may intersect at: ∅, vertex, edge, facet.
The triangulation is unique for generic inputs, i.e. no d + 2 sites lieon same hypersphere, i.e. every d + 1 sites define unique simplex.A Delaunay facet belongs to: exactly one simplex iff it belongs toCH(P), otherwise belongs to exactly two (neighboring) simplices.
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Complexity in general dimension
I Delaunay triangulation in Rd ' convex hull in Rd+1.
I Convex Hull of n points in Rd is Θ(n log n + nbd/2c)Hence d-Del = Θ(n log n + ndd/2e)
I Lower bound [McMullen] on space Complexity
I optimal deterministic [Chazelle], randomized [Seidel]algorithms
Optimal algorithms by lift/project: R2: Θ(n log n), R3: Θ(n2).
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Generalized constructions
In R2: Various geometric graphs defined on P are subgraphs ofDT (P), e.g. Euclidean minimum spanning tree (EMST) of P.
Delaunay triangulation DT (P) of pointset P ⊂ Rd : triangulations.t. no site in P lies in the hypersphere inscribing any simplex ofDT (P).
I DT (P) contains d-dimensional simplices.
I hypersphere = circum-hypersphere of simplex.
I DT (P) is unique for generic inputs, i.e. no d + 2 sites lie onthe same hypersphere, i.e. every d + 1 sites define uniqueDelaunay “triangle”.
I Rd : Delaunay facet belongs to exactly one simplex ⇔ belongsto CH(P)
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Plane Decomposition Representation
• Doubly Connected Edge List (DCEL)– stores: vertices, edges and cells (faces);– for every (undirected) edge: 2 twins (directed) half-edges.
• Space complexity: O(|V |+ |E |+ n),|V | = #vertices, |E | = #edges, n = #input sites.– v : O(1): coordinates, pointer to half-edge where v is starting.– half-e O(1): start v , right cell, pointer next/previous/twin half-e
• DCEL operations:– Given cell c , edge e ⊂ c , find (neighboring) cell c ′: e ⊂ c ′: O(1)– Given cell, print every edge of cell: O(|E |).– Given vertex v find all incident edges: O(#neighbors).