Barycentric coordinates

Within traditional nsage there is no way in which to define the coordinates of a fictitious ideal point, in the same way as for all other points on a line. 

The procedure which is commonly used to establish the centre of gravity between two weighted points, by the lever rule of mechanics, suggests a new way of defining coordinates that would apply to all points, without exception (Coxeter, 1989). 

Two weights, ti and 2, suspended at the points A and B are balanced on a support placed at P, the centre of gravity, on condition that 

This balance is not disturbed when both weights are increased by the same 

Irrespective of the measured positions of points A and B on the line, the correct condition of balance will always be described by assigning coordinates (fi, 2) to point P, where AB = ti + 2- Alternatively AB = fiti + /J,t2, with coordinates /iti, fj,t2) for point P. The coordinates of point P is independent of measure and conveniently formulated as  

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Each component may therefore be viewed as having barycentric coordinates, and any choice of a specific set of barycentric coordinates is the design of a scheme. 

We can look on any binary scheme as an affine combination of B-splines, and choosing the barycentric coordinates in such a combination is a very convenient viewpoint for designing a scheme to have specific properties. 

Once the dimension of the space to be considered has been reduced to a small enough number, the ranges of the barycentric coordinates for which other properties take acceptable values can be determined by a simple search. We are aided in this by the fact that most of these properties are continuous with respect to the c, and so a relatively coarse sampling is adequate to give a good idea. 

By the same procedure barycentric coordinates can be set up in the plane of a reference triangle Ai A2A3. If p -bp +P 0 masses p, p, p at the three vertices determine a point P (the centroid) with coordinates (P,P,P). 

In projective geometry the line at infinity no longer has a special role and barycentric coordinates may be replaced by general projective coordinates... 

Barycentric coordinates can be referred to any given triangle with vertices (1,0,0), (0,1,0), (0,0,1) and imit point (1,1,1), the centroid. In contrast, projective coordinates can be applied to any quadrangle Take three of the four vertices to determine a system of barycentric coordinates and suppose that the fourth vertex is (/Ui,/U2> Ms)- Converted to projective coordinates the fourth vertex becomes (1,1,1). Whereas all triangles are alike in affine geometry, ali quadrangles are seen to be alike in projective geometry. 

These polynomials Aj, A2, A3 are said to be the barycentric coordinates of the sgtencil [Pg.343]

Due to the scale-invariance of the barycentric coordinates Ai, A2, A3, the condition number of the matrix, arising from the expansion (6), is independent of the local mesh width h, see [1]. We remark that this observation is very crucial when working with adaptive mesh refinement, where very small cells may appear. Indeed, the representation (6) is, due to its robustness, particularly suited for adaptive mesh refinement, even for strongly distorted meshes. 

Theorists calculate cross sections in the CM frame while experimentalists usually measure cross sections in the laboratory frame of reference. The laboratory (Lab) system is the coordinate frame in which the target particle B is at rest before the collision i.e. Vg = 0. The centre of mass (CM) system (or barycentric system) is the coordinate frame in which the CM is at rest, i.e. v = 0. Since each scattering of projectile A into (v[i, (ji) is accompanied by a recoil of target B into (it - i[/, ([) + n) in the CM frame, the cross sections for scattering of A and B are related by... 

FIGURE 8.5 Relationship between the barycentric and Descartes coordinate systems, two-dimensional example. 

The origin of the coordinates is the barycentre of the environment of the considered atom and the eigenvectors Vi, V2> Vs associated with the eigenvalues are unit vectors which define the three principal axes of this environment. The considered ith atom can be included or not in its environment. To each atom of the environment is assigned an atomic property pj (e.g. unitary property, atomic mass, atomic electronegativity, atomic van der Waals volume) and a weight wj which is a function of the distance of the /th environment atom from the / th atom. 

The barycentric v of the vertices from 0 to N — 1 is calculated through the arithmetic mean of their coordinates by excluding the worst vertex v. ... 

We use the following simplified version of the skeleton, assuming a barycentric placement in space, emphasizing the substitutable positions by bullets and introducing cartesian coordinates ... 

Figure 5. FD ion velocity distributions shown in Cartesian coordinates in the center of mass system for barycentric energies (a) 0.21 eV, (b) 0.39 eV, and (c) 0.91 eV. "+), system s velocity of the center of mass (X), most probable velocity of the ideal spectator stripping product at 0°. (0), most probable values of Q = —0.20, —0.26, —0.55 for (a), (b), and (c), repsectively (Q), the spectator stripping values of Q —0.12, —0.21, —0.50 for (a), (b), and (c), respectively. The single contour lines to the right of each product distribution correspond to the 50% intensity profile of the primary ion beam.

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