I was thinking about this today. And although I've made some trials to answer it, none of them seem to fit. So, how do we define interior and exterior of a geometric figure? I guess that perhaps there might be some connection with the idea of open/closed sets in topology. But I'm not sure.
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Wasn't this asked recently, possibly on another SE? – HDE 226868 Dec 03 '14 at 23:03
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@HDE: Yep, What makes the inside of a shape the inside? – Dec 04 '14 at 05:10
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What seems relevant to me is the Jordan curve theorem (http://en.wikipedia.org/wiki/Jordan_curve_theorem ):
"In topology, a Jordan curve is a non-self-intersecting continuous loop in the plane, and another name for a Jordan curve is a simple closed curve. The Jordan curve theorem asserts that every Jordan curve divides the plane into an "interior" region bounded by the curve and an "exterior" region containing all of the nearby and far away exterior points, so that any continuous path connecting a point of one region to a point of the other intersects with that loop somewhere."
So, once you have a point that is either inside or outside a closed. non-intersecting curve, all points that can be connected to that point without intersecting the curve are also either inside or outside the curve.
marty cohen
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From a computer graphics standpoint, there is an easy to understand odd parity rule. Here's a link: http://en.wikipedia.org/wiki/Even%E2%80%93odd_rule. Another rule is the "winding" number rule, a little harder to grasp. The same link will take you to a short discussion of this rule. – johng Dec 04 '14 at 05:56