C H A P T E R 3
Geometry, Topology, and Shape
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
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