A convenient way to represent
is obtained by *identification*, which is a general
method of declaring that some points of a space are identical, even
though they originally were distinct.^{4.6} For a
topological space , let denote that has been redefined
through some form of identification. The open sets of become
redefined. Using identification,
can be defined as
, in which the identification declares that 0 and
are equivalent, denoted as . This has the effect of
``gluing'' the ends of the interval together, forming a closed loop.
To see the homeomorphism that makes this possible, use polar
coordinates to obtain
. You should already be familiar with 0 and leading
to the same point in polar coordinates; here they are just normalized
to 0 and . Letting run from 0 up to , and then
``wrapping around'' to 0 is a convenient way to represent
because
it does not need to be curved as in (4.5).

It might appear that identifications are cheating because the definition of a manifold requires it to be a subset of . This is not a problem because Whitney's theorem, as mentioned previously, states that any -dimensional manifold can be embedded in . The identifications just reduce the number of dimensions needed for visualization. They are also convenient in the implementation of motion planning algorithms.

Steven M LaValle 2020-08-14