> I would like relate
> these to the polylines returned by the Google Directions API so that I
> can display a map that color-codes the Google polylines.

I'll have a stab that the question generalises to:
Given an arbritary polyline (the Directions), you want to test if that
overlies, in whole or in part, some of your own collection of
arbritary polylines.
I guess the expected results are a bunch of your polylines that
together make up the Directions line.
I don't think that will be at all easy.
Start here
http://www.google.com/search?q=test+matching+polylines

I cannot see a good approach.
You can certainly get a set of candidates easily, by finding all the
suspects whose bounds intersect the target directions bounds.  But
then it gets difficult.
I'm not sure if finding all lines that intersect, and then comparing
headings is useful.  That should at least allow discarding some
easily.  It _might_ be useful, as only a limited number of real roads
exist near each other without intersecting - but see dual carriageways
below.

Some factors you might need to consider are
"how close is close enough" - a few metres? The polylines are unlikely
to ever be an exact match.
"what about dual carriageways" - will you treat one side as
interchangeable with the other, or distinguish the directions of
travel?  There may other multilane circumstances, e.g. where an access
road lies alongside a major route but does not connect to it.
"do I want to find lines that only partially match" e.g. if your
dataset includes A-B-C and the directions is B-C-D does that match for
your purpose?

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