Fitting a general simple polygon to a regular polygon Is there a method for finding the best fit approximation of a general simple polygon (the coastline of metropolitan France, say) by a regular polygon with given number of sides (a hexagon, say), using any reasonable interpretation of "best fit"?

 A: I've done this kind of thing in the past.
The main idea is to use
the fit of a straight line
to a set of point
that is independent of any axis.
This is done by
the distance being minimized
being the sum of the squares
of the lines from the points
with the lines being
perpendicular to the
line being fitted.
I gave a method to do this
in an answer somewhere here - 
I'll see if I can find it.
Aha! Here's the link:
linear least squares minimizing distance from points to rays - is it possible?
Anyway,
once you have this method,
use an incremental method
to construct the
fitted lines.
Start with two consecutive points.
For each point adjacent to
the current points,
append the neighboring points.
If the fitted line for these points
(each considered separately)
has a mean square error that is small enough
(that's for you to decide),
keep that point,
and keep adding is that direction.
When the adding stops
in both directions,
that is one of your lines.
Start with two points in one of the
directions,
only,
from now on,
only move in that direction.
Continue this
until all the points are used.
The time for this
is linear in the number of points,
so you can try various experiments
such as starting at various initial points.
Good luck.
