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A circle of radius r rolls without slipping inside an n-gon of side length l. A curve C is traced out by a pencil through a hole a distance d from the centre. Initially the circle is in a corner with the line from the hole to the circle centre at an angle x to the line from that corner to the n-gon centre.
What is the area of the curve? A point is defined as inside C if any curve from that point to a point at infinity, which does not go through any points where C intersects itself, intersects C an odd number of times.

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You mean the area of the set of points interior to $C$, according to your definition. A point could be surrounded several times by $C$ - are you asking for area with multiplicities? –  user20266 Dec 31 '11 at 17:24
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While these links don't answer your exact question (you seem to be asking for the area of the set of points whose winding number is odd with respect to the curve), they appear to be related, so I wonder if you've seen them: http://www.mamikon.com/USArticles/GenCycloGons.pdf and http://www.mamikon.com/USArticles/CycloidAreas.pdf

Also, for the sake of pedantry, it might worth pointing out that not all radii will result in a closed curve as the problem is currently set up.

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