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Trilinear coordinates
In geometry, the trilinear coordinates ''x:y:z'' of a point relative to a given triangle describe the relative directed distances from the three sidelines of the triangle. Trilinear coordinates are an example of homogeneous coordinates. They are often called simply "trilinears". The ratio ''x:y'' is the ratio of the perpendicular distances from the point to the sides (extended if necessary) opposite vertices ''A'' and ''B'' respectively; the ratio ''y:z'' is the ratio of the perpendicular distances from the point to the sidelines opposite vertices ''B'' and ''C'' respectively; and likewise for ''z:x'' and vertices ''C'' and ''A''.
In the diagram at right, the trilinear coordinates of the indicated interior point are the actual distances (''a' '', ''b' '', ''c' ''), or equivalently in ratio form, ''ka' '':''kb' '':''kc' '' for any positive constant ''k''. If a point is on a sideline of the reference triangle, its corresponding trilinear coordinate is 0. If an exterior point is on the opposite side of a sideline from the interior of the triangle, its trilinear coordinate associated with that sideline is negative. It is impossible for all three trilinears to be non-positive.
==Notation==
The ratio notation ''x'':''y'':''z'' for trilinears is different from the ordered triple notation (''a' '', ''b' '', ''c' '') for actual directed distances. Here each of ''x'', ''y'', and ''z'' has no meaning by itself; its ratio to one of the others ''does'' have meaning. Thus "comma notation" for trilinears should be avoided, because the notation (''x'', ''y'', ''z''), which means an ordered triple, does not allow, for example, (''x'', ''y'', ''z'') = (2''x'', 2''y'', 2''z''), whereas the "colon notation" does allow ''x'' : ''y'' : ''z'' = 2''x'' : 2''y'' : 2''z''.
抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』
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