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Computational Topology (Jeff Erickson)
Examples of Cell Complexes
Arithmétique! algèbre! géométrie! trinité grandiose! triangle lumineux! Celui
qui ne vous a pas connues est un insensé! Il mériterait l’épreuve des plus grands
supplices; car, il ya du mépris aveugle dans son insouciance ignorante.
...
[Arithmetic! Algebra! Geometry! Grandiose trinity! Luminous triangle! Whoever
has not known you is a fool! He deserves the most intense torture, for there is
blind contempt in his ignorant indifference.
...]
— Le Comte de Lautréamont [Isidore Lucien Ducasse],
Chant Deuxième,
Les Chants de Maldoror
(1869)
À bas Euclide! Mort aux triangles! [Down with Euclid! Death to triangles!]
— Jean Dieudonné, keynote address at the Royaumont Seminar (1959)
15 Examples of Cell Complexes
15.1 Proximity Complexes of Point Clouds
Point clouds are an increasingly common representation for complex geometric objects or domains. For
many applications, instead of storing an explicit description of the domain, either because the object is
too complex or because it is simply unknown, it may be sufﬁcient to store a representative sample of
points from the object. Typical sources of pointcould data are scanners (such as digital cameras, laser
rangeﬁnders, LIDAR, medical imaging systems, and telescopes), edge and featuredetection algorithms
from computer vision, locations of sensors and adhoc network devices, Monte Carlo sampling and
integration algorithms, and training data for machine learning systems.
By themselves, point clouds have no interesting topology. However, there are several natural ways to
impose topological structure onto a point cloud, intuitively by ‘connecting’ points that are sufﬁciently
‘close’. If the underlying domain is sufﬁciently ‘nice’ and the point sample is sufﬁciently ‘dense’, we can
recover important topological features of the underlying domain.
15.1.1 Aleksandrov
ˇ
Cech Complexes: Nerves and Unions
Let
P
be a set of points in some metric space
S
, and let
"
be a positive real number. Typically, but
not universally, the point set
P
is ﬁnite and the underlying space
S
is the Euclidean space
R
d
. The
Aleksandrov
ˇ
Cech complex A
ˇ
C
"
(
P
)
is the
intersection complex
or
nerve
of the set of balls of radius
"
centered at points in
P
. That is,
k
+
1 points in
P
deﬁne a
k
simplex in
A
ˇ
C
"
(
P
)
if and only if the
"
balls
centered at those points have a nonempty common intersection, or equivalently, if those points lie
inside a ball of radius
"
. Formally, the Aleksandrov
ˇ
Cech complex is an
abstract
simplicial complex; its
simplices can overlap arbitrarily and can have arbitrarily high dimension. Aleksandrov
ˇ
Cech complexes
were developed independently by Pavel Aleksandrov
[
6
]
and Eduard
ˇ
Cech
[
16
]
1
; despite Aleksandrov’s
earlier work, they are more commonly known as
ˇ
Cech complexes
.
The Aleksandrov
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This note was uploaded on 01/24/2012 for the course CS 598 taught by Professor Staff during the Fall '08 term at University of Illinois, Urbana Champaign.
 Fall '08
 Staff

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