### 4.8 Distances

To deﬁne the distance between two objects is a concept universally graspable on an intuitive level. In
mathematical terms the notion of distance as reinforced by our every day experiences can be
expressed using the simple notion of an aﬃne space (Deﬁnition 79) and the lengths of the
vectors inhabiting the vector space, which is part of the aﬃne space’s deﬁnition. This simple
notion fails, however, when considering distances between geographical locations, since the
shape of the world can not be modelled as an aﬃne space, but necessarily takes the shape
of a manifold (Deﬁnition 37), even if this manifold is embeddable into an aﬃne space.

A simple example of this circumstance is presented in Figure 4.7. The manifold of a circle is embedded
in the aﬃne space of the plane. Considering the simple case, where the circumference of a circle equals
, the maximum distance between any two points on the circle is , while in contrast the
embedded case has a maximum distance of a mere .

Therefore a more abstract and general approach is presented in order to alleviate this issue. Before
the length can be assessed, it should be noted that the question whether any two points are
connected or not, is a question of topology (Deﬁnition 28), not of the geometry built on top of
this topology. The connection established by topology manifests itself in the existence of
curves (Deﬁnition 45) connecting the two points under consideration. As the number of
curves

joining the points and is inﬁnite, even when the set of points involved is unique, since diﬀerent
parametrization deﬁne diﬀerent curves, an additional selection criterion, the length of a curve , which
is independent of the parametrization, is required to select a particular curve and assign the distance
. The length of a segment of a curve may be deﬁned using integration (Deﬁnition 95) along
the curve.
The distance is now readily obtained as the result of a minimization of the lengths of all the available
curves as described in Equation 4.176 connecting the desired points

As
this formulation uses the inﬁmum, it allows for the simple inclusion of open sets, which would
otherwise pose a problem. Applying this deﬁnition allows not only for the proper treatment of distances
within Riemannian manifolds (Deﬁnition 78), which enable proper treatment of distances along the
globe, but also encompasses settings used in the theory of relativity [4]. In these cases the term is
connected to the metric tensor ﬁeld (Deﬁnition 76).