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DIMAp – UFRN
[email protected]
Prof. Marcelo Ferreira Siqueira
Summer School on Computational Geometry - 2013, IMPA, Rio de
Janeiro, RJ, Brazil
Lecture 3
Computational Geometry:
Delaunay Triangulations and
Voronoi Diagrams
2
Triangulations
Given a finite family, (ai)i⇥I , of points in En, we say
that(ai)i⇥I is affinely independent if and only if the family of
vec-tors, (aiaj)j⇥(I�{i}), is linearly independent for some i ⇥
I.
a0a1
a2a1
E2 E2 E2
a0a0
A. I. A. I.A. I.
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Triangulations
Given a finite family, (ai)i⇥I , of points in En, we say
that(ai)i⇥I is affinely independent if and only if the family of
vec-tors, (aiaj)j⇥(I�{i}), is linearly independent for some i ⇥
I.
Not A. I.
a1
a2a1
a2
a3
E2 E2
a0
a0
Not A. I.
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4
Triangulations
In En, the largest number of affinely independent points is
n + 1.
Let a0
, . . . , ad be any d + 1 affinely independent points inEn.
a0a1
a2a1
E2 E2 E2
a0a0
5
Triangulations
The simplex s spanned by the points a0
, . . . , a
d
is the convex
hull, conv({a0
, . . . , a
d
}), of these points, and is denoted by[a
0
, . . . , a
d
].
The points a0
, . . . , ad are the vertices of s.
a0a1
a2a1
E2 E2 E2
a0a0
6
Triangulations
a0a1
a2a1
E2 E2 E2
a0
a0
The simplex s spanned by the points a0
, . . . , a
d
is the convex
hull, conv({a0
, . . . , a
d
}), of these points, and is denoted by[a
0
, . . . , a
d
].
The dimension, dim(s), of s is d, and s is called a
d-simplex.
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7
Triangulations
a0a1
a2a1
E2 E2 E2
a0a0
The simplex s spanned by the points a0
, . . . , a
d
is the convex
hull, conv({a0
, . . . , a
d
}), of these points, and is denoted by[a
0
, . . . , a
d
].
In En, we have simplices of dimension 0, 1, . . . , n only.
8
Triangulations
A 0-simplex is a point, a 1-simplex is a line segment, a 2-
simplex is a triangle, and a 3-simplex is a tetrahedron, and
so on.
0-simplex
3-simplex2-simplex
1-simplex
9
Triangulations
The convex hull of any nonempty (proper) subset of ver-
tices of a simplex s is also a simplex, called a (proper)
face
of s.
a0 a1
a2
a 2-face: [a0, a1, a2]
a0 a1
a2
3 proper 0-faces: [a0
], [a1
], [a2
]
a0 a1
a2
3 proper 1-faces: [a0
, a1
], [a0
, a2
], [a1
, a2
]
a0 a1
a2
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10
Triangulations
A simplicial complex, K, in En is a finite set of simplices inEn
such that
(1) if s ⇥ K and t � s then t ⇥ K, and
(2) if s ⌅ t ⇤= ∆ then s ⌅ t � s, t, for all s, t ⇥ K,
where a � b denotes "a is a (not necessarily proper) face
ofb".
11
Triangulations
violates (1) violates (2)
A simplicial complex
12
Triangulations
The dimension, dim(K), of a simplicial complex K is thelargest
dimension of a simplex in K. We refer to a d-dimensional simplicial
complex as simply a d-(simplicial)
complex.
a 2-complex
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13
Triangulations
Note that a simplicial complex is a discrete object (i.e.,
afinite collection of simplices). In turn, each simplex is a
set
of points.
a 2-complex
14
Triangulations
The (point) set consisting of the union of all points in the
simplices of a simplicial complex, K, is called the underly-ing
space of K and denoted by |K|. Note that K is a contin-uous
object.
its underlying space
15
Triangulations
E2
5 points in E2 convex hull of the 5 points
A triangulation of a nonempty and finite set, P, of points
of
En, is a simplicial complex, T (P), such that all vertices ofT
(P) are in P and the union of all simplices of T (P)
equalsconv(P).
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16
Triangulations
E2
two triangulations of the same point set
A triangulation of a nonempty and finite set, P, of points
of
En, is a simplicial complex, T (P), such that all vertices ofT
(P) are in P and the union of all simplices of T (P)
equalsconv(P).
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Triangulations
E2
P conv(P) T (P)
In our definition we do assume that the affine hull of Phas
dimension n. So, a triangulation of P may contain nosimplex of
dimension n, such as the example below forn = 2:
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Triangulations
Note that not all points of P need to be vertices of T (P),
ex-cept for the extremes points of conv(P), which are alwaysin T
(P).
Whenever all points in P are vertices of T (P), we call T (P)a
full-triangulation. This is the type of triangulation we will
study.
From now on, we drop the word "full" and refer to the term
"triangulation" as a full-triangulation of the given point
set.
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19
Triangulations
(proof discussed in the end of the lecture)
Theorem 3.1. Every nonempty and finite set, P ⇢ E2, ad-mits a
triangulation, which partitions the convex hull of
P.
20
The Delaunay Triangulation
Let P be a nonempty and finite set of points in E2.
E2
co-circularity
E2
collinearity
For the time being, let us assume that (1) not all pointsof P
are collinear, and (2) no four points of P lie in thecircumference
defined by 3 of them. Observe that "(1) )|P| � 3".
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The Delaunay Triangulation
The Lifting Procedure
Let w : E2 ! R be the function defined as
w(p) = x2 + y2 ,
for every p = (x, y) 2 E2.
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22
The Delaunay Triangulation
Note that w can be seen as a height function that lifts the
point p = (x, y) to the paraboloid of equation z = x2 + y2
in E3.
p = (x, y)
w(p)
E3
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The Delaunay Triangulation
p = (x, y)
w(p)
E3
LetP
w = {(x, y, w(x, y)) � E3 | (x, y) � P} .
(x, y, w(x, y))
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The Delaunay Triangulation
Note that P is the orthogonal projection of P
wonto the xy-
plane.
p = (x, y)
w(p)
E3
(x, y, w(x, y))
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25
The Delaunay Triangulation
E3
Consider the convex hull, conv(Pw), of Pw. Denote it byP .
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The Delaunay Triangulation
Consider the convex hull, conv(Pw), of Pw. Denote it byP .
P
E3
27
The Delaunay Triangulation
Project the lower envelope of P onto the xy-plane.
P
orthogonally
E3
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E3
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The Delaunay Triangulation
Project the lower envelope of P onto the xy-plane.
P
If the result of the projection of the lower envelope of P is
atriangulation, then we call it the Delaunay triangulation of
P.
E2
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The Delaunay Triangulation
We denote the Delaunay triangulation of P by DT (P).
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The Delaunay Triangulation
Proposition 3.2. Let S ⇢ E3 be the paraboloid given bythe
equation z = x2 + y2, and let H ⇢ E3 be a non-verticalhyperplane,
i.e., one whose normal vector has non-zero
last coordinate. Let C be the projection of H \ S into E2
obtained by dropping the last coordinate of all points in
H \ S. Then, C is either empty, a single point, or a
circum-ference.
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31
The Delaunay Triangulation
(proof on the board)
x
z
y
C
HS
Proposition 3.2.
32
The Delaunay Triangulation
Lemma 3.3. Let Q ⇢ P be any subset of P with 3
affinelyindependent points. Then, Qw corresponds to the vertexset
of a lower facet of the polytope, P , if and only if allpoints in Q
lie on a circle and all points of P�Q are outsideit.
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The Delaunay Triangulation
x
z
y
S
C
H
(proof on the board)
Lemma 3.3.
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34
The Delaunay Triangulation
By hypothesis, no four points of P lie in the same
circum-ference.
So, Lemma 3.3 implies that all facets of the lower envelope
of P are triangles in E3, and so are their projections
ontoE2.
By hypothesis, not all points of P are collinear. This meansthat
the proper faces of the lower envelope of P are trian-gles.
35
The Delaunay Triangulation
What can we conclude from the previous remarks?
The projection of the lower envelope of P is a set of
trian-gles.
It turns out that — with a little bit of an effort — we can
also
show that this set of triangles, along with their edges and
vertices, is a triangulation of P. This implies that the
edges
and vertices of the triangles must be in the lower envelope
of P .
36
The Delaunay Triangulation
By definition, the triangulation resulting from the projec-
tion of the lower envelope is the Delaunay triangulation,
DT (P).
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37
The Delaunay Triangulation
(proof on the board)
Lemma 3.4. Let P be a nonempty and finite set of pointsof E2. If
no four points of P lie in the same circumferenceand not all points
of P lie in the same line, then the Delau-nay triangulation, DT
(P), of P exists. Furthermore, it isunique.
38
The Delaunay Triangulation
What if one these two assumptions does not hold?
E2
co-circularity
E2
collinearity
39
The Delaunay Triangulation
E3
If all points of P are collinear, then P is 2-dimensionaland its
lower envelope is a polygonal chain containing all
points of P.
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The projection of the lower envelope is also a polygonal
chain containing all points of P, which is a triangulation
aswell.
40
The Delaunay Triangulation
According to our definition!
41
The Delaunay Triangulation
From our definition of Delaunay triangulation, this "de-
generate" triangulation is also the Delaunay triangulation
of P.
42
The Delaunay Triangulation
Suppose that all points in Q lie in the same
circumference,C.
E3
Let Q be a subset of P containing at least four points.
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43
The Delaunay Triangulation
E3
The points in Q cannot be all collinear. Otherwise, theywould
not lie in the same circumference (assuming "finite"radius).
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The Delaunay Triangulation
E3
If there is no point of P inside C, then Lemma 3.3 also tellsus
that Q is exactly the vertex set of a lower envelope facetof P
.
45
The Delaunay Triangulation
E3
So, the projection of the lower envelope is not a triangula-
tion.
The projection of this facet is a convex set with � 4
ver-tices!
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46
The Delaunay Triangulation
We call the resulting projection the Delaunay subdivision of
P.
Two-dimensional convex sets that are not triangles can al-
ways be triangulated (it is a well-known result in mathemat-
ics).
So, we can always obtain a triangulation of P from the De-launay
subdivision by triangulating those 2D convex sets.
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The Delaunay Triangulation
In practice, the resulting triangulation is usually called a
Delaunay triangulation. But, conceptually, this is not quite
right!
However, keep in mind that there is more than one wayof
"refining" a Delaunay subdivision to obtain a triangula-tion.
So, the triangulation (whatever we call it) is not unique!
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The Delaunay Triangulation
Let us summarize all facts we have learned so far...
Let P be any subset of points of E2.
If no four points of P lie in the same circumference, thenwe
know that the Delaunay triangulation, DT (P), of P ex-ists.
Furthermore, this triangulation contains no triangles if the
points of P are all collinear. Note that the converse
alsoholds.
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49
The Delaunay Triangulation
If four points of P lie in the same circumference and this
circumference contains no other point of P in its interior,
then
DT (P) does not exist, but the Delaunay subdivision of
Pdoes!
From any Delaunay subdivision of P, we can obtain a
tri-angulation of P by triangulating convex sets with � 4
ver-tices.
50
The Delaunay Triangulation
However, if for every circumference defined by (at least)four
points of P on it, there is always at least one point ofP inside
it, then the Delaunay triangulation, DT (P), of Pexists.
Why?
Because the lifting of the points on the circumference to
the
paraboloid doesn’t define a facet of the lower envelope of
P .
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The Delaunay Triangulation
What can we conclude from all these facts?
If P is nonempty and finite set of points of E2 such thatno four
points of P define a circumference whose interioris empty of points
of P, then DT (P) always exists and isunique.
Otherwise, we get the Delaunay subdivision, which canalways be
refined to yield a triangulation of P in a non-unique way.
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52
The Delaunay Triangulation
Theorem 3.1. Every nonempty and finite set, P ⇢ E2, ad-mits a
triangulation, which partitions the convex hull of
P.
We have a proof for Theorem 3.1:
We can also show its veracity by first proving the existence
of Delaunay triangulations and subdivisions of P ⇢ En,for n �
3.
The assertion of Theorem 3.1 also holds in En.
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The Delaunay Triangulation
We defined the Delaunay triangulation in E2 only, al-
though our definition of triangulation holds in En, for any
n 2 N.
So, our definition of DT (P) is the same for P � En, withn ⇥
N.
The lifting procedure can be extended to En, for n = 1 andn �
3.
54
The Delaunay Triangulation
If all points in P � En lie in the same hyperplane in En,then DT
(P) has no simplex of dimension n (like we sawfor n = 2).
If n + 2 points in P ⇢ En lie in the same sphere in En
and this sphere contains no point of P in its interior, thenthe
projection of the lower envelope of conv(P) is not aDelaunay
triangulation, but the Delaunay subdivision of
P.
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55
The Delaunay Triangulation
However, we can always refine the Delaunay subdivisionof P to
obtain a Delaunay triangulation of P, which is notunique.
We rely on the fact that any n-dimensional polytope,which is the
projection of a n-dimensional, lower facet ofconv(P) containing
more than n + 1 vertices, can be trian-gulated.
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The Delaunay Triangulation
Can you describe it?
This lemma gives us an algorithm for computing DT (P).
Consider Lemma 3.3 again:
Lemma 3.3. Let Q ⇢ P be any subset of P with 3
affinelyindependent points. Then, Qw corresponds to the vertexset
of a lower facet of the polytope, P , if and only if allpoints in Q
lie on a circle and all points of P�Q are outsideit.