In fact, the shape should not be that strange at all. It's just a half-circle per intersection point, is it not?
I suppose you could use an in-rectangle test (rectangle around the two intersection points, expanded by the radius of the test sphere) combined with a not-in-circle test (around each of the two intersection points) to build a somewhat simple in-corridor test.
Combined with some early out tests, that should be quite efficient.

Regards, Marius

On Wed, May 26, 2010 at 4:58 PM, <Paul_Firth@scee.net> wrote:
-----BEGIN PGP SIGNED MESSAGE-----
Hash: SHA256

> Assuming you have a choice in the method, would it not be easier to
> use the interior of any circles as walkable space, rather than the
> unions, and construct 'portals' between the line formed between the
> intersection points of each overlapping circle? Alternatively to
> reduce the number of circles significantly, circles can represent
> walkable space, circles can be explicitly linked together, and it
> could be assumed that the corridor built between the circles along
> their tangents is valid walkable space as well.

Hi Jeremy

I think the trouble with this idea is the shape of the corridors - in
actual fact, they're not rectangular they're strange curvy funnel shapes.
I'd be worried that simplifying assumption would lead to problems :)

Cheers, Paul.
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