Using the previously given conversion between barycentric and trilinear coordinates, the various other equations given in Trilinear coordinates # Formulas can be rewritten in terms of barycentric coordinates.
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However, in the situations where the important points of the studied problem are affinity independent, barycentric coordinates may lead to simpler computation, as in the following example.
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Instead, it is often easier to make a change of variables to any two barycentric coordinates, e . g . \ lambda _ 1, \ lambda _ 2.
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J . M . Tienstra ( 1895-1951 ) was a professor of the Delft university of Technology where he taught the use of barycentric coordinates in solving the resection problem.
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The barycentric coordinates define an affine isomorphism between the affine space and the affine subspace of defined by the equation \ lambda _ 0 + \ cdots + \ lambda _ n = 1.
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In particular, if a point lies on the opposite side of a sideline from the vertex opposite that sideline, then that point's barycentric coordinate corresponding to that vertex is negative.
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This is again a linear transformation, and we may extend the above procedure for triangles to find the barycentric coordinates of a point \ mathbf { r } with respect to a tetrahedron:
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That is, the Cartesian coordinates of any point are a weighted average of the Cartesian coordinates of the triangle's vertices, with the weights being the point's barycentric coordinates summing to unity.
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Thus if and only if two of the vertex angles differ from each other by more than 90? one of the barycentric coordinates is negative and so the nine-point center is outside the triangle.
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This result follows from the fact that a rectangle in barycentric coordinates corresponds to a quadrilateral in cartesian coordinates, and the ratio of the areas of the corresponding shapes in the corresponding coordinate systems is given by 2A.
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