Barycentric coordinate system
Therefore, barycentric coordinates are either considered to be defined up to multiplication by a nonzero constant, or normalized for summing to unity.
[7] Sometimes, they are also called affine coordinates, although this term refers commonly to a slightly different concept.
When working over the real numbers (the above definition is also used for affine spaces over an arbitrary field), the points whose all normalized barycentric coordinates are nonnegative form the convex hull of
The main advantage of barycentric coordinate systems is to be symmetric with respect to the n + 1 defining points.
On the other hand, distances and angles are difficult to express in general barycentric coordinate systems, and when they are involved, it is generally simpler to use a Cartesian coordinate system.
However this relationship is more subtle than in the case of affine coordinates, and, for being clearly understood, requires a coordinate-free definition of the projective completion of an affine space, and a definition of a projective frame.
[8] When constructing the projective completion from an affine coordinate system, one commonly defines it with respect to a projective frame consisting of the intersections with the hyperplane at infinity of the coordinate axes, the origin of the affine space, and the point that has all its affine coordinates equal to one.
Barycentric or areal coordinates are extremely useful in engineering applications involving triangular subdomains.
These make analytic integrals often easier to evaluate, and Gaussian quadrature tables are often presented in terms of area coordinates.
as vectors, so it makes sense to add or subtract them and multiply them by scalars.
are also called areal coordinates because they represent ratios of signed areas of triangles: One may prove these ratio formulas based on the facts that a triangle is half of a parallelogram, and the area of a parallelogram is easy to compute using a determinant.
Plug that into the previous line to obtain Therefore Similar calculations prove the other two formulas
Another way to solve the conversion from Cartesian to barycentric coordinates is to write the relation in the matrix form
The area interpretation of the barycentric coordinates can be recovered by applying Cramer's rule to this linear system.
The barycentric coordinates of a point can be calculated based on distances di to the three triangle vertices by solving the equation
If the point is not inside the triangle, then we can still use the formulas above to compute the barycentric coordinates.
If a point lies on an edge of the triangle but not at a vertex, one of the area coordinates
, then the point lies in that triangle or on its edge (explained in the previous section).
The integral of a function over the domain of the triangle can be annoying to compute in a cartesian coordinate system.
In the homogeneous barycentric coordinate system defined with respect to a triangle
This is again a linear transformation, and we may extend the above procedure for triangles to find the barycentric coordinates of a point
Once again, the problem of finding the barycentric coordinates has been reduced to inverting a 3×3 matrix.
Tetrahedral meshes are often used in finite element analysis because the use of barycentric coordinates can greatly simplify 3D interpolation.
As for the case of a simplex, the points with nonnegative normalized generalized coordinates (
) the generalized barycentric coordinates of a point are not unique, as the defining linear system (here for n=2)
Various kinds of additional restrictions can be used to define unique barycentric coordinates.
[15] More abstractly, generalized barycentric coordinates express a convex polytope with n vertices, regardless of dimension, as the image of the standard
Dual to generalized barycentric coordinates are slack variables, which measure by how much margin a point satisfies the linear constraints, and gives an embedding
This map is one-to-one (slack variables are uniquely determined) but not onto (not all combinations can be realized).
Generalized barycentric coordinates have applications in computer graphics and more specifically in geometric modelling.
