# barycentric coordinates

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## barycentric coordinates

(ba-ră-sen -trik) See barycenter.

## Barycentric Coordinates

the coordinates of a point M on a plane in relation to three basis points A1, A2, and A3 (not lying on the same line) of this plane—three numbers m1, m2, and m3 (which satisfy the condition m1 + m2 + m3 = 1), such that the point M represents the center of gravity of the system of three material points with masses m1, m2, and m3 located at the points A1, A2, and A3 respectively. (Here it is necessary to consider that the masses m1, m2, and m3 can be both positive and negative.) Barycentric coordinates in space are defined analogously. Barycentric coordinates are used in certain branches of mathematics and its applications.

## barycentric coordinates

[‚bar·ə′sen·trik kō′ȯrd·ən‚əts]
(mathematics)
The coefficients in the representation of a point in a simplex as a linear combination of the vertices of the simplex.
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In our approach, we utilize Barycentric coordinates to identify the corresponding points that map the geometric structure of the MS[S.sup.sim] to that of the MS[S.sup.ref] to define the behavioral correlation.
Then, the absolute values of the barycentric coordinates [a.sub.kl], [a.sub.km], and [a.sub.kn] can be computed as
To investigate the effect of the Re number on the variation of the barycentric coordinates, four different Re numbers, namely, 0.10, 0.15, 0.20, and 0.25, were considered.
A linear interpolation is used on tetrahedra for the soft tissue model, the displacement ut of a tissue constraint point placed inside a tetrahedron is given by the barycentric coordinates and the displacement [DELTA]qt of the 4 nodes ut = Jt[DELTA]qt.
By using the barycentric coordinates we determine next the vertex position on the original target surface.
For a 0-form, barycentric coordinates themselves constitute the suitable interpolant, that is
The barycentric coordinates [[mu].sub.i] corresponding to each [p.sub.i] are defined by the following properties: for each simplex [tau] in K,
It is assumed that P(x, y) is a point on triangle [T.sub.123] and its barycentric coordinates is denoted as ([l.sub.1], [l.sub.2], [l.sub.3]).
Finally, in order to prove (2.8), we recall that the basis function of the Crouzeix-Raviart elements associated with the midpoint of a side l can be written as [[lambda].sub.1] + [[lambda].sub.2] - [[lambda].sub.3], where [[lambda].sub.1] and [[lambda].sub.2] are the barycentric coordinates corresponding to the vertices of l and [[lambda].sub.3] is that corresponding to the opposite vertex.
The barycentric coordinates of a point x [element of] [Sigma] are real numbers [[Phi].sub.i] with
The keypoints on the mesh and their barycentric coordinates are extracted from a reference image in which the surface is in front of the camera without deformations.

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