Paths

Let be a topological space, which for our purposes will also be a
manifold. A *path* is a continuous function,
. Alternatively,
may be used for the domain of
. Keep in mind that a path is a function, not a set of points.
Each point along the path is given by for some
. This makes it appear as a nice generalization to the sequence
of states visited when a plan from Chapter 2 is
applied. Recall that there, a countable set of stages was defined,
and the states visited could be represented as , , .
In the current setting is used, in which replaces the
stage index. To make the connection clearer, we could use instead
of to obtain for each
.

Steven M LaValle 2012-04-20