Editing Barycentric coordinate system (section)

== Generalized barycentric coordinates ==
Barycentric coordinates <math>(\lambda_1, \lambda_2, ..., \lambda_k)</math> of a point <math>p \in \mathbb{R}^n</math> that are defined with respect to a finite set of ''k'' points <math>x_1, x_2, ..., x_k \in \mathbb{R}^n</math> instead of a [[simplex]] are called '''generalized barycentric coordinates'''. For these, the equation

<math display=block>(\lambda_1 + \lambda_2 + \cdots + \lambda_k)p = \lambda_1 x_1 + \lambda_2 x_2 + \cdots + \lambda_k x_k</math>

is still required to hold.<ref>{{Cite journal |last1=Meyer |first1=Mark |last2=Barr |first2=Alan |last3=Lee |first3=Haeyoung |last4=Desbrun |first4=Mathieu |date=6 April 2012 |title=Generalized Barycentric Coordinates on Irregular Polygons |url=http://www.geometry.caltech.edu/pubs/MHBD02.pdf |journal=Journal of Graphics Tools |volume=7 |pages=13–22 |doi=10.1080/10867651.2002.10487551|s2cid=13370238 }}</ref> Usually one uses normalized coordinates, <math>\lambda_1 + \lambda_2 + \cdots + \lambda_k = 1</math>. As for the case of a simplex, the points with nonnegative normalized generalized coordinates (<math>0 \le \lambda_i \le 1</math>) form the [[convex hull]] of {{math|''x''<sub>1</sub>, ..., ''x''<sub>''n''</sub>}}. If there are more points than in a full simplex (<math>k > n + 1</math>) the generalized barycentric coordinates of a point are ''not'' unique, as the defining linear system (here for n=2)<math display="block">
\left(\begin{matrix}
1 & 1 & 1 & ... \\
x_1 & x_2 & x_3 & ... \\
y_1 & y_2 & y_3 & ...
\end{matrix}\right)
\begin{pmatrix}
\lambda_1 \\ \lambda_2 \\ \lambda_3 \\ \vdots
\end{pmatrix} =
\left(\begin{matrix}
1\\x\\y
\end{matrix}\right)
</math>is [[Underdetermined system|underdetermined]]. The simplest example is a [[quadrilateral]] in the plane. Various kinds of additional restrictions can be used to define unique barycentric coordinates.<ref>{{Cite journal|last=Floater|first=Michael S.|date=2015|title=Generalized barycentric coordinates and applications *|url=https://www.mn.uio.no/math/english/people/aca/michaelf/papers/gbc.pdf|journal=Acta Numerica|language=en|volume=24|pages=161–214|doi=10.1017/S0962492914000129|s2cid=62811364 |issn=0962-4929}}</ref>

=== Abstraction ===
More abstractly, generalized barycentric coordinates express a convex polytope with ''n'' vertices, regardless of dimension, as the ''image'' of the standard <math>(n-1)</math>-simplex, which has ''n'' vertices – the map is onto: <math>\Delta^{n-1} \twoheadrightarrow P.</math> 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 except when P is a simplex.

[[Dual linear program|Dual]] to generalized barycentric coordinates are [[slack variable]]s, which measure by how much margin a point satisfies the linear constraints, and gives an [[embedding]] <math>P \hookrightarrow (\mathbf{R}_{\geq 0})^f</math> into the ''f''-[[orthant]], where ''f'' is the number of faces (dual to the vertices). This map is one-to-one (slack variables are uniquely determined) but not onto (not all combinations can be realized).

This use of the standard <math>(n-1)</math>-simplex and ''f''-orthant as standard objects that map to a polytope or that a polytope maps into should be contrasted with the use of the standard vector space <math>K^n</math> as the standard object for vector spaces, and the standard [[affine hyperplane]] <math>\{(x_0,\ldots,x_n) \mid \sum x_i = 1\} \subset K^{n+1}</math> as the standard object for affine spaces, where in each case choosing a [[linear basis]] or [[affine basis]] provides an ''isomorphism,'' allowing all vector spaces and affine spaces to be thought of in terms of these standard spaces, rather than an onto or one-to-one map (not every polytope is a simplex). Further, the ''n''-orthant is the standard object that maps ''to'' cones.

=== Applications ===
[[File:Barycentric RGB.svg|alt=Barycentric coordinates are used for blending three colors over a triangular region evenly in computer graphics.|thumb|Barycentric coordinates are used for blending three colors over a triangular region evenly in computer graphics.]]
Generalized barycentric coordinates have applications in [[computer graphics]] and more specifically in [[geometric model]]ling.<ref>{{Cite journal |last=Floater |first=Michael S. |date=2003 |title=Mean value coordinates |url=https://linkinghub.elsevier.com/retrieve/pii/S0167839603000025 |journal=Computer Aided Geometric Design |language=en |volume=20 |issue=1 |pages=19–27 |doi=10.1016/S0167-8396(03)00002-5}}</ref> Often, a three-dimensional model can be approximated by a polyhedron such that the generalized barycentric coordinates with respect to that polyhedron have a geometric meaning. In this way, the processing of the model can be simplified by using these meaningful coordinates. Barycentric coordinates are also used in [[geophysics]].<ref>ONUFRIEV, VG; DENISIK, SA; FERRONSKY, VI,  BARICENTRIC MODELS IN ISOTOPE STUDIES OF NATURAL-WATERS. NUCLEAR GEOPHYSICS, 4, 111-117 (1990)</ref>