Given is a surface patch embedded in $R^3$ with disk topology, i.e. it has a single boundary. I would like to know if it is possible to determine if the boundary is "interior" or "exterior" - an intuitive definition is given below, but can it be made mathematically rigorous?
For example, take a sphere that is cut in two by a plane (the two parts need not have equal size). The boundary of the larger part I would classify as "interior", since it could be seen as a hole in the surface (just think of an extreme case in which only a small cap is removed). The boundary of the smaller part would be "exterior". If the sphere is cut exactly along the equator, we get the indeterminate case.
Now, imagine you're walking on the surface along the boundary, so that the surface always lies to your right. Intuitively, the boundary should be classified as "interior" if you make more left turns than right turns on a complete loop, and vice versa.
Is there a way to rigorously define this notion?
I would actually like to have a computational criterion that distinguishes the cases. One idea, which I do not know how to formulate explicitly, would be to integrate along the boundary, summing up the deviations from the straight line in the tangent plane, e.g. a curve to the left counting positively, a curve to the right negatively. The sign of the integral over the entire contour would decide on the "orientation" (sorry I don't have a better term to describe this).