# 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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We use the standard equation to translate a barycentric coordinate to cartesian coordinate, which we can then use within our estimation algorithm to drive the behavior of the MS[S.sup.sim] to behave like the MS[S.sup.ref].
The barycentric coordinate of the rigid body is [[-36.7749, -6.2296, -208.3247].sup.T] in [o.sub.1]-[x.sub.1][y.sub.1][z.sub.1].
In 1827, the barycentric coordinate was developed to characterize the relative position of a point with respect to other points [23-26].
We also denote [p.sub.1][p.sub.2][p.sub.3] to be the face containing [p.sup.*] and denote [alpha], [beta], gamma] to be the barycentric coordinate of [p.sup.*] with respect to [p.sub.1][p.sub.2][p.sub.3]; that is, [p.sup.*] = [alpha][p.sub.1] + [beta][p.sub.2] + [gamma[p.sub.3].
Given a set of affine points u0, [u.sub.1], ..., [u.sub.p], p = 0, ..., 4 in 4-d spacetime, a p-simplex [[sigma].sub.p] is the region spanned by u = [[summation].sup.p.sub.i=0] [[lambda].sub.i][u.sub.i] for all possible values of [[lambda].sub.i] [member of] [0,1], where [[lambda].sub.i] is the barycentric coordinate associated with [u.sub.i] such that [[summation].sup.p.sub.i=0] [[lambda].sub.i] = 1.
Let [[mu].sub.i] define the barycentric coordinate corresponding to the i-th vartex [p.sub.i] of K as follows:
If a triangle with the barycentric coordinate as the initial point can translate freely along any straight line where a edge of the triangle lies, then the triangle is deemed to be impending.
The oscillation can be minimized in most cases by using polar splines or splines based on barycentric coordinate systems.
1) The moments of the universal mean time UT1 must be expressed in TT (Terrestrial Time), TCG (Geocentric Coordinate Time) and TCB (Barycentric Coordinate Time) time scales.
In 1991 we also gained the relativity-based Geocentric Coordinate Time (TCG) and Barycentric Coordinate Time (TCB), paralleling TT and TDB.
Assuming that a keypoint in the mesh in[x.sub.i], it can be expressed in terms of its barycentric coordinate of the facet where this keypoint lies on, as:
where (s,t) is the Barycentric coordinate of (x,y) in a triangle, and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] are the Lagrangian interpolation positions in the triangle given by

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