Barycentric coordinates

Coordinates of a point in an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015280/b0152801.png" />-dimensional vector space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015280/b0152802.png" />, with respect to some fixed system <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015280/b0152803.png" /> of points that do not lie in an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015280/b0152804.png" />-dimensional subspace. Every point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015280/b0152805.png" /> can uniquely be written as

<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015280/b0152806.png" />

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015280/b0152807.png" /> are real numbers satisfying the condition <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015280/b0152808.png" />. The point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015280/b0152809.png" /> is by definition the centre of gravity of the masses <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015280/b01528010.png" /> located at the points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015280/b01528011.png" />. The numbers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015280/b01528012.png" /> are called the barycentric coordinates of the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015280/b01528013.png" />; the point with barycentric coordinates <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015280/b01528014.png" /> is called the barycentre. Barycentric coordinates were introduced by A.F. Möbius in 1827, [1], as an answer to the question about the masses to be placed at the vertices of a triangle so that a given point is the centre of gravity of these masses. Barycentric coordinates are a special case of homogeneous coordinates; they are affine invariants.

Barycentric coordinates of a simplex are used in algebraic topology [2]. Barycentric coordinates of a point of an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015280/b01528015.png" />-dimensional simplex <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015280/b01528016.png" /> with respect to its vertices <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015280/b01528017.png" /> is the name given to its (ordinary) Cartesian coordinates in the basis of the vectors <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015280/b01528018.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015280/b01528019.png" /> is any point that does not lie in the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015280/b01528020.png" />-dimensional subspace carrying <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015280/b01528021.png" /> (if it is considered that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015280/b01528022.png" /> lies in some Euclidean space, then the definition does not depend on the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015280/b01528023.png" />), or to projective coordinates with respect to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015280/b01528024.png" /> in the projective completion of the subspace containing <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015280/b01528025.png" />. The barycentric coordinates of the points of a simplex are non-negative and their sum is equal to one. If the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015280/b01528026.png" />-th barycentric coordinate becomes zero, this means that the point lies at the side of the simplex <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015280/b01528027.png" /> opposite to the vertex <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015280/b01528028.png" />. This makes it possible to consider the barycentric coordinates of the points of a geometric complex with respect to all of its vertices. Barycentric coordinates are used to construct the barycentric subdivision of a complex.

Barycentric coordinates of abstract complexes are formally defined in an analogous manner [3].

References

[1]	A.F. Möbius, "Der barycentrische Kalkul" , Gesammelte Werke , 1 , Hirzel , Leipzig (1885)
[2]	L.S. Pontryagin, "Grundzüge der kombinatorischen Topologie" , Deutsch. Verlag Wissenschaft. (1956) (Translated from Russian)
[3]	E.H. Spanier, "Algebraic topology" , McGraw-Hill (1966)