Since the Cartesian coordinates of any point are a weighted average of those of the vertices, with the weights being the point's barycentric coordinates normalized to sum to unity, the circumcenter vector can be written as
42.
The first step is to find the barycentric coordinates of the origin, and then the test that the simplex contains the origin is simply that all the barycentric coordinates are positive and the sum is less than 1.
43.
The first step is to find the barycentric coordinates of the origin, and then the test that the simplex contains the origin is simply that all the barycentric coordinates are positive and the sum is less than 1.
44.
If the are viewed as bodies that have weights ( or masses ) \ lambda _ i, the point is thus the barycenter of the, and this explains the origin of the term " barycentric coordinates ".
45.
Slack variables are " dual " to generalized barycentric coordinates, and, dually to generalized barycentric coordinates ( which are not unique but can all be realized ), are uniquely determined, but cannot all be realized.
46.
Slack variables are " dual " to generalized barycentric coordinates, and, dually to generalized barycentric coordinates ( which are not unique but can all be realized ), are uniquely determined, but cannot all be realized.
47.
The map is one-to-one if and only if the polytope is a simplex, in which case the map is an isomorphism; this corresponds to a point not having " unique " generalized barycentric coordinates.
48.
To find the reverse transformation, from Cartesian coordinates to barycentric coordinates, we first substitute \ lambda _ { 3 } = 1-\ lambda _ { 1 }-\ lambda _ { 2 } \, into the above to obtain
49.
Explicitly, the formulae for the barycentric coordinates of point \ mathbf { r } in terms of its Cartesian coordinates ( " x, y " ) and in terms of the Cartesian coordinates of the triangle's vertices are:
50.
Consequently, even normalized generalized barycentric coordinates ( i . e . coordinates such that the sum of the coefficients is 1 ) are in general not uniquely determined anymore while this is the case for normalized barycentric coordinates with respect to a simplex.
