Computational Geometry and Its Application to GIS 129
Proof: The proof comes about by computing the sums of the signed area
of the triangles defined by any point in the plane (e.g. the origin) and the
edges of the polygon, and expanding all the determinants given above, and
then simplifying.
This theorem can also be generalized for computing volumes, where the
term in the summation is over the determinants of the simplicial faces (e.g.
triangles in 3-space) of the boundary of the polyhedron.
Note that in low dimensional space (2 and 3) these determinants can be
computed by simply expanding their definitions. In higher dimensions it is
usually better to compute determinants by converting it into upper triangular
form by, say, Gauss elimination method,