The total signed area of the projection of an oriented surface onto an oriented flat plane depends only on the boundary and not on any of the interior points. The "shadow" does not change if the interior points are moved around. If the surface is deformed so that there is an "overhang" where some projected points fall outside of the projected boundary, such as in the example of the right, these points cancel out with the points on the opposite side (top or bottom) of the overhang. An upside down surface projects negative area, and in the example on the right, all negative projected area is cancelled out with the positive area projected by the upright surface on top of the overhang.
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