# area bounded by spirogram

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