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C H A P T E R 3

Geometry, Topology, and Shape

Representation

In this chapter, we introduce a number of concepts, focusing on those which are common to

most of the following chapters. Many more concepts will be introduced later on, to support the

formalization of advanced techniques for shape analysis.

We begin with the deﬁnition of metric spaces, that is, spaces where a function, a metric, is

deﬁned to formalize the notion of distance between points. e intuition we have of distance is

rooted in the 3D Euclidean space we live in, where the distance between two points is the length

of the straight line that connects them. Yet there are other ways to measure distances: imagine

you are given two points on the boundary of a 3D object and you are asked to measure the length

of the shortest path one would follow if bounded to walk on the boundary of the object itself.

is way to measure distances yields, in general, a diﬀerent result than measuring along a straight

line, and it generalizes the Euclidean distance to geodesic distances in curved spaces.

Topological space is another basic mathematical concept which serves to model shapes and

generalizes to arbitrary spaces concepts we are familiar with, as they are proper to the Euclidean

space we live in. ese concepts include, for example, those of closeness, connectedness, continuity.

Building on topological spaces, a term we will encounter several times throughout the book is

manifold: manifolds are the mathematical expression of spaces with a well-behaved structure and

smoothness degree.

Continuity is among the ﬁrst concepts we learn in basic mathematical courses and is an

example of well-behaved mapping between a domain and its co-domain. is notion may be

extended further to deﬁne the theoretical framework within which two spaces, or shapes, might

be considered equivalent. Functions between topological spaces of a certain type allow us to consider

two shapes as if they were the same: this is a powerful technique as we may want to perform the

analysis not in its original shape space but in some transformed space, where computations or

reasoning might be simpler.

3.1 METRIC AND METRIC SPACES

We are going to start our mathematical journey introducing the notion of metric, which has to

do with a basic idea in human experience, namely, the idea of distance. In everyday life, the

term distance means some degree of closeness of two physical objects or concepts (e.g., in space

or time), and the term metric usually stands for a measurement. e mathematical meaning of

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