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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