Oriented Area: Doubly-Covered Regions

Just as a region may have either negative or positive area depending on the orientation of its boundary curve, it is possible for a region to be counted more than once towards the algebraic area of a polygon.

It may not be immediately clear why the central pentagonal region in figure 5(a) should be given any more weight than the triangular regions in the star. Whether you consider the "Star of David" in figure 5(b) as two triangles or as one disjoint "hexagon," you expect to get the same result for its area. However, when considered as two triangles, the inner hexagonal region clearly must be counted twice. This double counting of regions happens as a consequence of triangular subdivision and the formula for triangles as described in the previous section, and allows many natural geometric interpretations.

This feature of oriented area is necessary for the geometric interpretation of the formula for the area of a midpoint pentagon, as well as for the general case. The central pentagonal region is counted once in the original polygon, but with a factor of 1/2; this same region is counted twice in the "star polygon," but with a factor of 1/4. These contributions sum to 1, as expected.

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