35 - 2
Poisson Delaunay triangulation
• Poisson distribution• Slivnyak-Mecke formula• Blaschke-Petkanschin variables substitution• Stupid analysis of the expected degree• Straight walk expected analysis• Catalog of properties
36 - 1
Poisson distribution
Distribution in A independent from distribution in B.
Unit uniform rate
when A \ B = ?
P [|X \A| = k] =vol(A)
k
k!e� vol(A)
X a Poisson point process
36 - 2
Poisson distribution
Distribution in A independent from distribution in B.
Unit uniform rate
when A \ B = ?
P [|X \A| = k] =vol(A)
k
k!e� vol(A)
V
e
r
y
c
o
n
v
e
n
i
e
n
t
X a Poisson point process
36 - 3
Poisson distribution
Distribution in A independent from distribution in B.
Unit uniform rate
when A \ B = ?
P [|X \A| = k] =vol(A)
k
k!e� vol(A)
E [|X \ A|] =1X
0
kvol(A)k
k!e� vol(A)
= vol(A)
P [|X \A| = 0] = e� vol(A)
X a Poisson point process
37 - 2
Slivnyak-Mecke formulaX a Poisson point process of density n
Sum Integral
E
2
4X
q2X
1[P (X,q)]
3
5
37 - 3
Slivnyak-Mecke formulaX a Poisson point process of density n
Sum Integral
E
2
4X
q2X
1[P (X,q)]
3
5= n
Z
R2
P [P (X \ {q}, q)] dq
37 - 4
Slivnyak-Mecke formulaX a Poisson point process of density n
Sum Integral
E
2
4X
q2X
1[P (X,q)]
3
5= n
Z
R2
P [P (X \ {q}, q)] dq
e.g.,
E
2
4X
q2X
1[NNX(0)=q]
3
5
37 - 5
Slivnyak-Mecke formulaX a Poisson point process of density n
Sum Integral
E
2
4X
q2X
1[P (X,q)]
3
5= n
Z
R2
P [P (X \ {q}, q)] dq
e.g.,
E
2
4X
q2X
1[NNX(0)=q]
3
5= n
Z
R2
P [D(0, kqk) \X = ;] dq
37 - 6
Slivnyak-Mecke formulaX a Poisson point process of density n
Sum Integral
E
2
4X
q2X
1[P (X,q)]
3
5= n
Z
R2
P [P (X \ {q}, q)] dq
e.g.,
E
2
4X
q2X
1[NNX(0)=q]
3
5= n
Z
R2
P [D(0, kqk) \X = ;] dq
= n
Z
R2
e�n⇡kqk2
dq
37 - 7
Slivnyak-Mecke formulaX a Poisson point process of density n
Sum Integral
E
2
4X
q2X
1[P (X,q)]
3
5= n
Z
R2
P [P (X \ {q}, q)] dq
e.g.,
E
2
4X
q2X
1[NNX(0)=q]
3
5= n
Z
R2
P [D(0, kqk) \X = ;] dq
= n
Z
R2
e�n⇡kqk2
dq
= n
Z2⇡
0
Z 1
0
e�n⇡r2rd✓ dr = n⇥ 2⇡ ⇥ 1
2n⇡ = 1
38 - 2
Blaschke-Petkantschin variable substitutionZ
(R2)
3
f(p, q, t) dp dq dt
=
Z 1
0
Z 1
�1
Z 1
�1
Z2⇡
0
Z2⇡
0
Z2⇡
0
f(p, q, t)|det(J)|d↵1
d↵2
d↵3
dxdydr
38 - 3
Blaschke-Petkantschin variable substitutionZ
(R2)
3
f(p, q, t) dp dq dt
=
Z 1
0
Z 1
�1
Z 1
�1
Z2⇡
0
Z2⇡
0
Z2⇡
0
f(p, q, t)|det(J)|d↵1
d↵2
d↵3
dxdydr
=
Z 1
0
Z 1
�1
Z 1
�1
Z2⇡
0
Z2⇡
0
Z2⇡
0
f(p, q, t)2r3area(↵1
↵2
↵3
)d↵1
d↵2
d↵3
dxdydr
(x, y) r
↵1
↵2
↵3
39 - 3
Expected number of triangles in conflict with origin
E
2
4 1
6
X
p,q,t2X3
1[pqr2DT (X)]
1[O2Disk(pqt)]
3
5
X a Poisson point process of density n
39 - 4
Expected number of triangles in conflict with origin
E
2
4 1
6
X
p,q,t2X3
1[pqr2DT (X)]
1[O2Disk(pqt)]
3
5
=
n3
6
Z
(R2)
3
P [X \B(pqt) = ;] 1[O2Disk(pqt)] dp dq dt
X a Poisson point process of density n
Slivnyak-Mecke formula
39 - 5
Expected number of triangles in conflict with origin
E
2
4 1
6
X
p,q,t2X3
1[pqr2DT (X)]
1[O2Disk(pqt)]
3
5
=
n3
6
Z
(R2)
3
P [X \B(pqt) = ;] 1[O2Disk(pqt)] dp dq dt
=
n3
6
Z 1
0
Z r
0
Z2⇡
0
Z2⇡
0
Z2⇡
0
Z2⇡
0
e�n⇡r22r3area(↵
1
↵2
↵3
)Rd↵1
d↵2
d↵3
d✓dRdr
X a Poisson point process of density n
Blaschke-Petkantschin formula
39 - 6
Expected number of triangles in conflict with origin
E
2
4 1
6
X
p,q,t2X3
1[pqr2DT (X)]
1[O2Disk(pqt)]
3
5
=
n3
6
Z
(R2)
3
P [X \B(pqt) = ;] 1[O2Disk(pqt)] dp dq dt
=
n3
6
Z 1
0
Z r
0
Z2⇡
0
Z2⇡
0
Z2⇡
0
Z2⇡
0
e�n⇡r22r3area(↵
1
↵2
↵3
)Rd↵1
d↵2
d↵3
d✓dRdr
=
n3
6
Z 1
0
e�n⇡r2r3ÇZ r
0
RdR
åÇZ2⇡
0
d✓dR
ådr
ÇZ2⇡
0
Z2⇡
0
Z2⇡
0
2area(↵1
↵2
↵3
)d↵1
d↵2
d↵3
å
X a Poisson point process of density n
39 - 7
Expected number of triangles in conflict with origin
E
2
4 1
6
X
p,q,t2X3
1[pqr2DT (X)]
1[O2Disk(pqt)]
3
5
=
n3
6
Z
(R2)
3
P [X \B(pqt) = ;] 1[O2Disk(pqt)] dp dq dt
=
n3
6
Z 1
0
Z r
0
Z2⇡
0
Z2⇡
0
Z2⇡
0
Z2⇡
0
e�n⇡r22r3area(↵
1
↵2
↵3
)Rd↵1
d↵2
d↵3
d✓dRdr
=
n3
6
Z 1
0
e�n⇡r2r3ÇZ r
0
RdR
åÇZ2⇡
0
d✓dR
ådr
ÇZ2⇡
0
Z2⇡
0
Z2⇡
0
2area(↵1
↵2
↵3
)d↵1
d↵2
d↵3
å
X a Poisson point process of density n
Maple computation:
> assume(n>0):with(LinearAlgebra):
> int( exp(-n*Pi*r^ 2)*r^ 5,r=0..infinity);
1/(n^ 3*Pi^ 3)
> 6*int(int(int(Determinant([[ 1, 1, 1],
[cos(alpha1),cos(alpha2),cos(alpha3)],
[sin(alpha1),sin(alpha2),sin(alpha3)]]),
alpha1=0..alpha2),alpha2=alpha3),alpha3=0..2*Pi);
24*Pi^ 2
(R, ✓)
r
↵1
↵2
↵3
39 - 8
Expected number of triangles in conflict with origin
E
2
4 1
6
X
p,q,t2X3
1[pqr2DT (X)]
1[O2Disk(pqt)]
3
5
=
n3
6
Z
(R2)
3
P [X \B(pqt) = ;] 1[O2Disk(pqt)] dp dq dt
=
n3
6
Z 1
0
Z r
0
Z2⇡
0
Z2⇡
0
Z2⇡
0
Z2⇡
0
e�n⇡r22r3area(↵
1
↵2
↵3
)Rd↵1
d↵2
d↵3
d✓dRdr
=
n3
6
Z 1
0
e�n⇡r2r3ÇZ r
0
RdR
åÇZ2⇡
0
d✓dR
ådr
ÇZ2⇡
0
Z2⇡
0
Z2⇡
0
2area(↵1
↵2
↵3
)d↵1
d↵2
d↵3
å
=
n3
6
Z 1
0
e�n⇡r2r32⇡r2
2
dr 24⇡2
=
n3
6
⇡1
n3⇡3
24⇡2
= 4
X a Poisson point process of density n
39 - 9
Expected number of triangles in conflict with origin
E
2
4 1
6
X
p,q,t2X3
1[pqr2DT (X)]
1[O2Disk(pqt)]
3
5
=
n3
6
Z
(R2)
3
P [X \B(pqt) = ;] 1[O2Disk(pqt)] dp dq dt
=
n3
6
Z 1
0
Z r
0
Z2⇡
0
Z2⇡
0
Z2⇡
0
Z2⇡
0
e�n⇡r22r3area(↵
1
↵2
↵3
)Rd↵1
d↵2
d↵3
d✓dRdr
=
n3
6
Z 1
0
e�n⇡r2r3ÇZ r
0
RdR
åÇZ2⇡
0
d✓dR
ådr
ÇZ2⇡
0
Z2⇡
0
Z2⇡
0
2area(↵1
↵2
↵3
)d↵1
d↵2
d↵3
å
=
n3
6
Z 1
0
e�n⇡r2r32⇡r2
2
dr 24⇡2
=
n3
6
⇡1
n3⇡3
24⇡2
= 4
) Eîd�DT (X\{0})(0)
ó= 6
(you already know )
X a Poisson point process of density n
40 - 4
Straight walk analysis
E
2
4 1
2
X
p,q,t2X3
1[pqt2DT (X)]
1[p below,q above]
1[pq intersects segment]
3
5X a Poisson point process of density n
40 - 5
Straight walk analysis
E
2
4 1
2
X
p,q,t2X3
1[pqt2DT (X)]
1[p below,q above]
1[pq intersects segment]
3
5X a Poisson point process of density n
= E
2
4 1
2
X
p,q,t2X3
1[pqt2DT (X)]
1[p below,q,t above]
1[pq intersects segment]
3
5
+E
2
4 1
2
X
p,q,t2X3
1[pqt2DT (X)]
1[p,t below,q above]
1[pq intersects segment]
3
5
= E
2
4X
p,q,t2X3
1[pqt2DT (X)]
1[p below,q,t above]
1[pq intersects segment]
3
5
40 - 6
Straight walk analysis
Slivnyak-Mecke formula
= n3
Z
(R2)
3
P [X \B(pqt) = ;] 1["position"] dp dq dt
X a Poisson point process of density n
E
2
4X
p,q,t2X3
1[pqt2DT (X)]
1[p below,q,t above]
1[pq intersects segment]
3
5
40 - 7
Straight walk analysis
= n3
Z
(R2)
3
P [X \B(pqt) = ;] 1["position"] dp dq dt
Blaschke-Petkantschin formula
· r32area(↵1
↵2
↵3
)d↵1
d↵2
d↵3
dxdydr
' n3
Z 1
0
Z1
0
Z r
�r
Z2⇡
0
Z2⇡
0
Z2⇡
0
e�n⇡r21["position"]
X a Poisson point process of density n
E
2
4X
p,q,t2X3
1[pqt2DT (X)]
1[p below,q,t above]
1[pq intersects segment]
3
5
40 - 8
Straight walk analysisX a Poisson point process of density n
· r32area(↵1
↵2
↵3
)d↵1
d↵2
d↵3
dxdydr
' n3
Z 1
0
Z1
0
Z r
�r
Z2⇡
0
Z2⇡
0
Z2⇡
0
e�n⇡r21["position"]
40 - 9
Straight walk analysisX a Poisson point process of density n
· r32area(↵1
↵2
↵3
)d↵1
d↵2
d↵3
dxdydr
' n3
Z 1
0
Z1
0
Z r
�r
Z2⇡
0
Z2⇡
0
Z2⇡
0
e�n⇡r21["position"]
· r32area(↵1
↵2
↵3
)d↵1
d↵2
d↵3
dydr
' n3
Z 1
0
Z r
�r
Z2⇡
0
Z2⇡
0
Z2⇡
0
e�n⇡r21["position"]
40 - 10
Straight walk analysisX a Poisson point process of density n
· r32area(↵1
↵2
↵3
)d↵1
d↵2
d↵3
dxdydr
' n3
Z 1
0
Z1
0
Z r
�r
Z2⇡
0
Z2⇡
0
Z2⇡
0
e�n⇡r21["position"]
· r32area(↵1
↵2
↵3
)d↵1
d↵2
d↵3
dydr
' n3
Z 1
0
Z r
�r
Z2⇡
0
Z2⇡
0
Z2⇡
0
e�n⇡r21["position"]
· r32area(↵1
↵2
↵3
)d↵3
d↵2
d↵1
rdhdr
' n3
Z 1
0
Z1
�1
Z2⇡�arcsinh
⇡+arcsinh
Z ⇡+arcsinh
� arcsinh
Z ⇡+arcsinh
� arcsinh
e�n⇡r2
rh = y
q,↵2
t,↵3
p,↵1
� arcsin
yr
40 - 11
Straight walk analysisX a Poisson point process of density n
· r32area(↵1
↵2
↵3
)d↵1
d↵2
d↵3
dxdydr
' n3
Z 1
0
Z1
0
Z r
�r
Z2⇡
0
Z2⇡
0
Z2⇡
0
e�n⇡r21["position"]
· r32area(↵1
↵2
↵3
)d↵3
d↵2
d↵1
rdhdr
' n3
Z 1
0
Z1
�1
Z2⇡�arcsinh
⇡+arcsinh
Z ⇡+arcsinh
� arcsinh
Z ⇡+arcsinh
� arcsinh
e�n⇡r2
' n3
Z 1
0
e�n⇡r2r3dr
⇥Z
1
�1
Z 2⇡�arcsinh
⇡+arcsinh
Z ⇡+arcsinh
� arcsinh
Z ⇡+arcsinh
� arcsinh
2area(↵1
↵2
↵3
)d↵3d↵2d↵1rdh
40 - 12
' n3
Z 1
0
e�n⇡r2r3dr
⇥Z
1
�1
Z 2⇡�arcsinh
⇡+arcsinh
Z ⇡+arcsinh
� arcsinh
Z ⇡+arcsinh
� arcsinh
2area(↵1
↵2
↵3
)d↵3d↵2d↵1rdh
Straight walk analysis
512
9
r
X a Poisson point process of density n
· r32area(↵1
↵2
↵3
)d↵1
d↵2
d↵3
dxdydr
' n3
Z 1
0
Z1
0
Z r
�r
Z2⇡
0
Z2⇡
0
Z2⇡
0
e�n⇡r21["position"]
' n3
Z 1
0
e�n⇡r2r3 dr
ask Maple !
40 - 13
' n3
Z 1
0
e�n⇡r2r3dr
⇥Z
1
�1
Z 2⇡�arcsinh
⇡+arcsinh
Z ⇡+arcsinh
� arcsinh
Z ⇡+arcsinh
� arcsinh
2area(↵1
↵2
↵3
)d↵3d↵2d↵1rdh
Straight walk analysis
E
2
4 1
2
X
p,q,t2X3
1[pqt2DT (X)]
1[p below,q above]
1[pq intersects segment]
3
5
512
9
r
X a Poisson point process of density n
' n3
Z 1
0
e�n⇡r2r3 dr
=
512
9
n3
3
8⇡2n2
pn=
64
3⇡2
pn ' 2.16
pn
512
9
r
42 - 4
Expected degree
E [(d
�(p)] = 62D
E [(d
�(p)] = 48⇡2
35
+ 2 ' 15.5353D
3D on a cylinder E [(d
�(p)] = ⇥(log n)
42 - 5
Expected degree
E [(d
�(p)] = 62D
E [(d
�(p)] = 48⇡2
35
+ 2 ' 15.5353D
3D on a cylinder E [(d
�(p)] = ⇥(log n)
3D on a surface O(1) E [(d
�(p)] O(log n)
c
o
n
j
e
c
t
u
r
egeneric
43 - 1
Expected maximum degree
Poisson distribution intensity 1, window [0,pn]2
E [max(d
�(p)] = ⇥
Älogn
log logn
ä
43 - 2
Expected maximum degree
Poisson distribution intensity 1, window [0,pn]2
E [max(d
�(p)] = ⇥
Älogn
log logn
ä
no boundaries!
43 - 3
Expected maximum degree
Poisson distribution intensity 1, window [0,pn]2
E [max(d
�(p)] = ⇥
Älogn
log logn
ä
Poisson distribution intensity n, bounded domain
E [max(d
�(p)] = O
�log
2+✏ n�
46 - 1
Shortest path
Upper pathVoronoi path
Shortened V. pathCompass walk
Expected length (experiments)
Euclidean length 11.041.071.161.181.27
Walk between vertices
46 - 2
Shortest path
Upper pathVoronoi path
Shortened V. pathCompass walk
Expected length (experiments)
Euclidean length 11.041.071.161.181.27
theory
� 1 + 10
�11
1.16n
u
m
e
r
i
c
a
l
i
n
t
e
g
r
a
t
i
o
n
35
3⇡2 ' 1.184
⇡ ' 1.27
[
B
a
c
c
e
l
l
i
e
t
a
l
.
,
2
0
0
0
]
Walk between vertices
47 - 4
Smoothed analysis of convex hull
K unit ball of Rd
initial point set
Add noise, uniform in �K
47 - 5
Smoothed analysis of convex hull
K unit ball of Rd
initial point set
Add noise, uniform in �K
Convex hull
47 - 7
Smoothed analysis of convex hull
K unit ball of Rd
special case: (✏,) sample
Add noise, uniform in �K
47 - 8
Smoothed analysis of convex hull
K unit ball of Rd
special case: (✏,) sample
Add noise, uniform in �K
Convex hull
47 - 9
Smoothed analysis of convex hull
1
2
0
1
3
1
Worst-case boundOur smoothed upper boundOur lower bound on regular n-gonLower bound from average-case
lim
n 1
log �
log n
lim
n 1
log bound
log n
1
4
2
3
-2
Dimension 2
2
3
10
33
� 1
2
OÄn
14 ��
38
ä
47 - 10
Smoothed analysis of convex hull
Open problems
Tighter analysis for CH
Delaunay size in 3D
Delaunay walk in 2D