Coordinate system based on distances from the sidelines of a given triangle
In geometry, the trilinear coordinatesx : 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. 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 trilinear coordinates to be non-positive.
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In geometry, the trilinearcoordinates x : y : z of a point relative to a given triangle describe the relative directed distances from the three sidelines...
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are the side-lengths of a reference triangle then the point whose trilinearcoordinates are f ( a , b , c ) : f ( b , c , a ) : f ( c , a , b ) {\displaystyle...
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therefore the ratio of weights at A and C being α : γ. In trilinearcoordinates, a point with coordinates x :y :z has perpendicular distances to side BC (across...
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