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Areal coordinate systems (aka barycentric coordinate systems) are used for some astrophysics purposes, e.g., some calculations involving strong-field gravity and general relativity. Its coordinates for a point consist of the areas of shapes that imply the point's position compared to a fixed set of points (of known position) that serve the typical role of origin and axes. For a two-dimensional areal coordinate system, three fixed points (forming a fixed triangle on the plane) serve this role. For any point within this triangle, three additional "sub triangles" are implicit, each consisting of this point and two of the larger triangle's vertices, i.e., sharing a side with the larger, defining triangle. This point's three areal coordinates are the ratios of the areas of each of these three smaller triangles with that of the defining triangle (the term areal is used as an adjective meaning "having to do with areas"). The coordinates must be ordered by a convention indicating which side of the defining triangle the coordinate corresponds to. Knowing any two of the coordinates is sufficient to determine the third and the point's location, since they are ratios that add to one. For a point outside the defining triangle, the triangles joining the defining triangle's sides to the point are larger and extend outside the defining triangle. In this case, if a coordinate's triangle is entirely outside the defining triangle, the coordinate is given a negative sign. Those with triangles partially within the defining triangle remain positive. The three coordinates still add to one.
For three-dimensional coordinates, a defining three-dimensional simplex (i.e., a 3-simplex aka tetrahedron) is used analogously to the defining triangle used in the two-dimensional case, a 3-simplex being a polyhedron of four faces, each of which is a triangle. A point within space is defined by four volume-ratios of four simplexes formed by the point and the defining simplex's four faces, in an analogous manner to the two-dimensional case. Simplexes of larger dimensions are analogous, and areal coordinates of larger dimension use them analogously.
The areal coordinate system is obscure, but it has properties that simplify some challenging calculations, and is used for some gravity calculations.