Editing Barycentric coordinate system (section)

==Relationship with projective coordinates==
Homogeneous barycentric coordinates are also strongly related with some [[projective coordinates]]. However this relationship is more subtle than in the case of affine coordinates, and, for being clearly understood, requires a coordinate-free definition of the [[projective completion]] of an [[affine space]], and a definition of a [[projective frame]].

The ''projective completion'' of an affine space of dimension {{mvar|n}} is a [[projective space]] of the same dimension that contains the affine space as the [[set complement|complement]] of a [[hyperplane]]. The projective completion is unique [[up to]] an [[isomorphism]]. The hyperplane is called the [[hyperplane at infinity]], and its points are the [[points at infinity]] of the affine space.<ref name="Berger">{{Citation | last1=Berger | first1=Marcel | author1-link=Marcel Berger | title=Geometry I | publisher=Springer | location=Berlin | isbn= 3-540-11658-3 | year=1987}}</ref>

Given a projective space of dimension {{mvar|n}}, a ''projective frame'' is an ordered set of {{math|''n'' + 2}} points that are not contained in the same hyperplane. A projective frame defines a projective coordinate system such that the coordinates of the {{math|(''n'' + 2)}}th point of the frame are all equal, and, otherwise, all coordinates of the {{mvar|i}}th point are zero, except the {{mvar|i}}th one.<ref name=Berger/>

When constructing the projective completion from an affine coordinate system, one commonly defines it with respect to a projective frame consisting of the intersections with the hyperplane at infinity of the [[coordinate axes]], the origin of the affine space, and the point that has all its affine coordinates equal to one. This implies that the points at infinity have their last coordinate equal to zero, and that the projective coordinates of a point of the affine space are obtained by completing its affine coordinates by one as {{math|(''n'' + 1)}}th coordinate.

When one has {{math|''n'' + 1}} points in an affine space that define a barycentric coordinate system, this is another projective frame of the projective completion that is convenient to choose. This frame consists of these points and their [[centroid]], that is the point that has all its barycentric coordinates equal. In this case, the homogeneous barycentric coordinates of a point in the affine space are the same as the projective coordinates of this point. A point is at infinity if and only if the sum of its coordinates is zero. This point is in the direction of the vector defined at the end of {{slink||Definition}}.