Lecture 1 Discrete Geometric Structures - Inria · Outline 1 Convex Polytopes 2 Voronoi diagrams and Delaunay triangulations 3 Weighted Voronoi diagrams 4 Union of balls 5 Discrete

Lecture 1Discrete Geometric Structures

Jean-Daniel Boissonnat

Winter School on Computational Geometry and TopologyUniversity of Nice Sophia Antipolis

January 23-27, 2017

Computational Geometry and Topology Discrete Geometric Structures J-D. Boissonnat 1 / 47

Outline

1 Convex Polytopes

2 Voronoi diagrams and Delaunay triangulations

3 Weighted Voronoi diagrams

4 Union of balls

5 Discrete metric spaces


Convex Polytopes

Two ways of defining convex polytopes


Intersection of n half-planes

Input : n half-planes Hi : aix+ biy + ci ≤ 0, i = 1, ..., n

Output : the convex polygon (possibly empty) H =⋂ni=1 Hi


Convex hull of n points

Input : n points p1, ..., pn in R2

Output : the convex polygon conv(P ) =∑n

i λipiλi ≥ 0∑n

i λi = 1

Equivalently, conv(P ) is the intersection of the halfplanes that

are bounded by lines passing through 2 points of P

contain all the points of P


Space of lines and duality

Duality point/line(non vertical) line h = (x, y) : y = ax− b −→ dual point h∗ = (a, b)

point p = (a, b) −→ dual line p∗ = (x, y) : y = ax− b.

The mapping ∗

is an involution and is thus bijective : h∗∗ = h and p∗∗ = p

preserves incidences :

p = (x, y) ∈ h⇐⇒ y = ax− b⇐⇒ b = xa− y ⇐⇒ h∗ ∈ p∗.

preserves inclusions

p ∈ h+ ⇐⇒ h∗ ∈ p∗+ (h+ = (x, y) : y > ax− b)


Intersections of halfplanes and convex hulls

Let h1, . . . , hn be n lines and let H = ∩h+i

ss

h1h2 *

h3

h∗3

h∗2

h∗1

A vertex s of H is the intersection point of two lines hi and hjlying above all other lines hk, k 6= i, j

=⇒ s∗ = (h∗ih∗j )

s∗ is a line supporting conv−(h∗i )

From a computational point of view, the two problems are equivalent


Facial structure of a polytope

Convex polytopes and duality can be defined in very much the same in anyfixed dimension

Supporting hyperplane h :H ∩ P 6= ∅P on one side of h

Faces : P ∩ h, h supp. hyp.

Dimension of a face :the dim. of its affine hull


General position

Points in general position

I P is in general position iff no subset of k + 2 points lie in a k-flat

⇒ any k-face is the convex hull of k + 1 point of P , i.e. a k-simplex

conv(P ) is called a simplicial polytope

Hyperplanes in general position

I H is in general position iff the intersection of any subset of d− khyperplanes intersect in a k-flat

⇒ any k-face is the intersection of d− k hyperplanes⋂H is called a simple polytope


1 Convex Polytopes



4 Union of balls



Voronoi diagrams in nature


The solar system (Descartes)


Growth of merystem


Euclidean Voronoi diagrams

Voronoi cell V (pi) = x : ‖x− pi‖ ≤ ‖x− pj‖, ∀j

Voronoi diagram (P ) = collection of all cells V (pi), pi ∈ P


Voronoi diagrams and convex polyhedra

Convex PolyhedronThe intersection of a finite collection of half-spaces : V =

⋂i∈I h

+i

Each Voronoi cell is a convex polyhedron

The Voronoi diagram has the structure of a cell complex: two cellseither do not intersect or intersect in a common face

The Voronoi diagram of P is the projection of a convex polyhedron ofRd+1



Vor(p1, . . . , pn) is the minimization diagramof the n functions δi(x) = (x− pi)2

argmin(δi) = argmax(hi)where hpi(x) = 2 pi · x− p2i

The minimization diagram of the δi is alsothe maximization diagram of the affinefunctions hpi(x)

The faces of Vor(P ) are the projections ofthe faces of V(P ) =

⋂i h

+pi

h+pi = x : xd+1 > 2pi · x− p2i

pi

z = (x− pi)2

Note !

the graph of hpi(x) is the hyperplane tangentto Q : xd+1 = x2 at (x, x2)



Lifting mapThe faces of Vor(P ) are the projection of the faces of the polytope

V(P ) =⋂i h

+pi

where hpi is the hyperplane tangent to paraboloid Q at the liftedpoint (pi, p

2i )

CorollariesI The size of Vor(P) is the same as the size of V(P )

I Computing Vor(P) reduces to computing V(P )


Voronoi diagram and Delaunay complex

Finite set of points P ∈ Rd

The Delaunay complex is the nerve of the Voronoi diagram

It is not always embedded in Rd


Empty circumballs

An (open) d-ball B circumscribing asimplex σ ⊂ P is called empty if

1 vert(σ) ⊂ ∂B2 B ∩ P = ∅

Del(P) is the collection of simplicesadmitting an empty circumball


Point sets in general position wrt spheres

P = p1, p2 . . . pn is said to be in general position wrt spheres if6 ∃ d+ 2 points of P lying on a same (d− 1)-sphere

Theorem [Delaunay 1936]

If P is in general position wrt spheres, Del(P ) has a natural realization inRd called the Delaunay triangulation of P .


Proof of Delaunay’s theorem 1

σ

h(σ)

PLinearization

S(x) = x2 − 2c · x+ s, s = c2 − r2

S(x) < 0 ⇔z < 2c · x− s (h−S )z = x2 (P)

⇔ x = (x, x2) ∈ h−S


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

⇔ ∀i, pi ∈ h+Sσ

⇔ σ is a face of conv−(P )

Del(P ) = proj(conv−(P ))


VD and DT in the space of spheres

hpi : xd+1 = 2pi · x− p2i pi = (pi, p

2i ) = h∗pi

V(P ) = h+p1 ∩ . . . ∩ h+pnduality−→ D(P ) = conv−(p1, . . . , pn)

↑ ↓Voronoi Diagram of P

nerve−→ Delaunay Complex of P


Voronoi diagrams, Delaunay triangulations and polytopes

If P is in general position wrt spheres :

V(P ) = h+p1 ∩ . . . ∩ h+

pn

duality−→ D(P ) = conv−(p1, . . . , pn)

↑ ↓

Voronoi Diagram of Pnerve−→ Delaunay Complex of P


1 Convex Polytopes



4 Union of balls



Foams, molecules and weighted Voronoi diagrams


Weighted Voronoi (or Power or Laguerre) Diagrams

Sites : n balls B = bi(pi, ri), i = 1, . . . nPower distance: π(x, bi) = (x− pi)2 − r2iIt is not a distance

Power Diagram: Vor(B)One cell V (bi) for each siteV (bi) = x : π(x, bi) ≤ π(x, bj).∀j 6= i

Each cell is a convex polyhedron

V (bi) may be empty

pi may not belong to V (bi)


Power diagrams are maximization diagrams

Cell of bi in the weighed Voronoi diagram Vor(B)

V (bi) = x ∈ Rd : π(x, bi) ≤ π(x, bj).∀j 6= i

= x ∈ Rd : 2pix− si = maxj∈[1,...n]2pjx− sjσ

h(σ)

P

Vor(B) is the maximization diagram of the set of affine functions

hi(x) = 2pix− si, i = 1, . . . , n

Geometrically : the vertical projection of P ∩ h−i is the ball bi


Weighted Delaunay triangulations

B = bi(pi, ri) a set of balls

Del(B) = nerve of Vor(B)

Let Bτ = bi(pi, ri), i = 0, . . . k ⊂ B :

Bτ ∈ Del(B)⇐⇒ ⋂bi∈Bτ V (bi) 6= ∅


Weighted VD and DT in the space of spheres

hbi : xd+1 = 2pi · x− p2i + r2

i φ(bi) = (pi, p2i − r2

i ) = h∗bi

V(B) = h+b1 ∩ . . . ∩ h+bn

duality−→ D(B) = conv−(φ(b1), . . . , φ(bn))↑ ↓

Voronoi Diagram of Bnerve−→ Delaunay Complex of B


1 Convex Polytopes



4 Union of balls



Union of balls

What is the combinatorial complexity of the boundary of the union Uof n balls of Rd ?

Compare with the complexity of the arrangement of the boundinghyperspheres

How can we compute U ?

What is the image of U in the space of spheres ?


Weighted Voronoi Diagram

σi(x) = (x− pi)2 − r2i


Restriction of Del(B) to U =⋃b∈B b

U =⋃b∈B b ∩ V (b) and ∂U ∩ ∂b = V (b) ∩ ∂b.

The nerve of C is the restriction of Del(B) to U , i.e. the subcomplexDel|U (B) of Del(B) whose faces have a circumcenter in U

∀b, b ∩ V (b) is convex and thus contractible

C = b ∩ V (b), b ∈ B is a good cover of U

The nerve of C is a deformation retract of Uhomotopy equivalent (Nerve theorem)


Interfaces between proteins [Cazals & Janin 2006]

Interface antigen-antibody


Fonctions distance et modeles de croissance


Mobius Diagrams

Wi = (pi, λi, µi)

δM (x,Wi) = λi‖x− pi‖2 − µibisectors are hyperspheres(hyperplanes or ∅)

Mob(Wi) = x, δ(x, σi) ≤ δ(x, σj)


Linearization Lemma

We can associate to each weighted point Wi

a hypersphere Σi of Rd+1 so that

the faces of the Mobius diagram of the Wi are obtained by

1 computing the Laguerre Diagram of the Σi

2 intersecting this diagram with the paraboloid P3 projecting vertically the faces of this intersection onto Rd


Proof

λi(x− pi)2 − µi ≤ λj(x− pj)2 − µj

⇐⇒ (x− λipi)2 + (x2 + λi2 )2 − λ2

i p2i −

λ2i4 + λip

2i − µi

≤ (x− λjpj)2 + (x2 +λj2 )2 − λ2

jp2j −

λ2j4 + λjp

2j − µj

⇐⇒ (X − Ci)2 − ρ2i ≤ (X − Cj)2 − ρ2

j

where X = (x, x2) ∈ Rd+1,

Ci = (λipi,−λi2 ) ∈ Rd+1 and ρ2

i = λ2i p

2i +

λ2i4 − λip2

i + µi

The Mobius diagram of the doubly weighted points Wi is the projectiononto Rd of the restriction to the paraboloid Q of the Laguerre diagram ofthe Σi = B(Ci, ρi)


1 Convex Polytopes



4 Union of balls



Discrete metric spaces : Witness Complex [de Silva]

L a finite set of points (landmarks) vertices of the complexW a dense sample (witnesses) pseudo circumcenters

The Witness ComplexWit(L, W )

w

Definition

2 Wit(L, W )()

8 9w 2 W withd(w, p) d(w, q)

8p 2 8q 2 L \

L : Landmarks (black dots); the vertices of the complexW : Witnesses (blue dots)

R. Dyer (INRIA) Del(L, M) = Wit(L, W ) Assisi, EuroCG 2012 2 / 10

Let σ be a (abstract) simplex with vertices in L, and let w ∈W . We saythat w is a witness of σ if

‖w − p‖ ≤ ‖w − q‖ ∀p ∈ σ and ∀q ∈ L \ σ

The witness complex Wit(L,W ) is the complex consisting of all simplexes σsuch that for any simplex τ ⊆ σ, τ has a witness in W


Easy consequences of the definition

The witness complex can be defined for any metric space and, inparticular, for discrete metric spaces

If W ′ ⊆W , then Wit(L,W ′) ⊆Wit(L,W )

Del(L) ⊆Wit(L,Rd)


Identity of Del|Ω(L) and Wit(L,Rd) [de Silva 2008]

Theorem : Wit(L,W ) ⊆Wit(L,Rd) = Del(L)

Remarks

I Faces of all dimensions have to be witnessed

I Wit(L,W ) is embedded in Rd if L is in general position wrt spheres


Proof of de Silva’s theorem

τ = [p0, ..., pk] is a k-simplex of Wit(L) witnessed by a ball Bτ (i.e. Bτ ∩ L = τ)We prove that τ ∈ Del(L) by induction on k

Clearly true for k = 0

Bσ

c

w

Bτ

τ

σ

Hyp. : true for k′ ≤ k − 1

B := Bτ

σ := ∂B ∩ τ , l := |σ|

// σ ∈ Del(L) by the hyp.

while l + 1 = dimσ < k do

B ← the ball centered on [cw] s.t.

- σ ⊂ ∂B,

- B witnesses τ

- |∂B ∩ τ | = l + 1

(B witnesses τ) ∧ (τ ⊂ ∂B) ⇒ τ ∈ Del(L)


Case of sampled domains : Wit(L,W ) 6= Del(L)

W a finite set of points ⊂ Ω ⊂ Rd

Wit(L,W ) 6= Del(L), even if W is a dense sample of Ω

ab

Vor(a, b)

[ab] ∈Wit(L,W ) ⇔ ∃p ∈W, Vor2(a, b) ∩W 6= ∅


Relaxed witness complex

Alpha-witness Let σ be a simplex with vertices in L. We say that a pointw ∈W is an α-witness of σ if

‖w − p‖ ≤ ‖w − q‖+ α ∀p ∈ σ and ∀q ∈ L \ σ

Alpha-relaxed witness complex The α-relaxed witness complexWitα(L,W ) is the maximal simplicial complex with vertex set L whosesimplices have an α-witness in W

Wit0(L,W ) = Wit(L,W )

Filtration : α ≤ β ⇒ Witα(L,W ) ⊆Witβ(L,W )


Interleaving complexes

Lemma Assume that W is ε-dense in Ω and let α ≥ 2ε. Then

Wit(L,W ) ⊆ Del|Ω(L) ⊆Witα(L,W )

Proofσ : a d-simplex of Del(L), cσ its circumcenterW ε-dense in Ω ∃w ∈W s.t. ‖cσ − w‖ ≤ εFor any p ∈ σ and q ∈ L \ σ, we then have

∀p ∈ σ and q ∈ L \ σ ‖w − p‖ ≤ ‖cσ − p‖+ ‖cσ − w‖≤ ‖cσ − q‖+ ‖cσ − w‖≤ ‖w − q‖+ 2‖cσ − w‖≤ ‖w − q‖+ 2ε


Documents

Lecture 1 Discrete Geometric Structures - Inria · Outline 1 Convex Polytopes 2 Voronoi diagrams and Delaunay triangulations 3 Weighted Voronoi diagrams 4 Union of balls 5 Discrete