Graphics Reference
In-Depth Information
C H A P T E R
7
Algebraic Topology and
Topology Invariants
Algebraic topology studies both topology spaces and functions through algebraic entities, such
as groups or homomorphisms, by analyzing the representations (formally known as
functors
) that
transform a topological problem into an algebraic one, with the aim of simplifying it. en, the
focus of algebraic topology is on the translation of the original problem into the algebraic lan-
guage.
A fundamental contribution of algebraic topology is its support to the formal definition of
a digital model and its description. On the one hand, the concepts of simplicial and cell complexes
support the definition of a computational representation scheme consistent with the mathematical
idealization of a shape; indeed cell decompositions are the most common geometric model used
in computer graphics and CAD/CAM [
134
,
147
]. On the other hand, homology groups analyze
and classify smooth manifolds and complexes; for this reason we think that homology offers a
set of mathematical tools (such as Betti numbers) that, possibly coupled with other descriptions,
yield a synthetic and expressive shape description. Homology is a powerful tool for shape analysis,
also because efficient algorithms for its computation exist. We refer the reader to well-known
textbooks such as [
99
,
135
,
149
,
173
] for a detailed treatment of other topics of algebraic topology.
7.1 CELL DECOMPOSITIONS
In order to construct topological spaces, one can take a collection of simple elements and glue
them together in a structured way. Probably the most relevant example of this construction is
given by simplicial complexes, whose building blocks are called simplices.
A
k
-
simplex
k
in
R
n
,
0kn
, is the convex hull of
kC1
affinely independent points
A
0
;A
1
;:::;A
k
, called
vertices
. Figure
7.1
shows the simplest examples of simplices:
0
is a point,
1
an interval,
2
a triangle (including its interior),
3
a tetrahedron (including its interior).
A
k
-simplex can be oriented by assigning an ordering to its vertices: two orderings of the
vertices that differ by an even permutation determine one and the same orientation of the
k
-
simplex. In this way, each
k
-simplex with
k > 0
can be given a positive or a negative orientation.
e
oriented
k
-simplex
with ordered vertices
.A
0
;A
1
;:::;A
k
/
is denoted by
A
0
;A
1
;:::;A
k
,
whereas the
k
-simplex with opposite orientation is denoted by
A
0
;A
1
;:::;A
k
.
A
face
of a
k
-simplex
k
is a simplex whose set of vertices is a non-empty subset of the set
of vertices of
k
. A
finite simplicial complex
can now be defined as a finite collection of simplices
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