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 2012-04-20