Barycentric coordinate system
In geometry, a barycentric coordinate system is a coordinate system in which the location of a point is specified by reference to a simplex. The barycentric coordinates of a point can be interpreted as masses placed at the vertices of the simplex, such that the point is the center of mass of these masses. These masses can be zero or negative; they are all positive if and only if the point is inside the simplex.
Every point has barycentric coordinates, and their sum is never zero. Two tuples of barycentric coordinates specify the same point if and only if they are proportional; that is to say, if one tuple can be obtained by multiplying the elements of the other tuple by the same non-zero number. Therefore, barycentric coordinates are either considered to be defined up to multiplication by a nonzero constant, or normalized for summing to unity.
Barycentric coordinates were introduced by August Möbius in 1827. They are special homogeneous coordinates. Barycentric coordinates are strongly related with Cartesian coordinates and, more generally, to affine coordinates.
Barycentric coordinates are particularly useful in triangle geometry for studying properties that do not depend on the angles of the triangle, such as Ceva's theorem, Routh's theorem, and Menelaus's theorem. In computer-aided design, they are useful for defining some kinds of Bézier surfaces.
Definition
Let be points in a Euclidean space, a flat or an affine space of dimension that are affinely independent; this means that there is no affine subspace of dimension that contains all the points, or, equivalently that the points define a simplex. Given any point there are scalars that are not all zero, such thatfor any point.
The elements of a tuple that satisfies this equation are called barycentric coordinates of with respect to The use of colons in the notation of the tuple means that barycentric coordinates are a sort of homogeneous coordinates, that is, the point is not changed if all coordinates are multiplied by the same nonzero constant. Moreover, the barycentric coordinates are also not changed if the auxiliary point, the origin, is changed.
The barycentric coordinates of a point are unique up to a scaling. That is, two tuples and are barycentric coordinates of the same point if and only if there is a nonzero scalar such that for every.
In some contexts, it is useful to constrain the barycentric coordinates of a point so that they are unique. This is usually achieved by imposing the condition
or equivalently by dividing every by the sum of all These specific barycentric coordinates are called normalized or absolute barycentric coordinates. Sometimes, they are also called affine coordinates, although this term refers commonly to a slightly different concept.
Sometimes, it is the normalized barycentric coordinates that are called barycentric coordinates. In this case the above defined coordinates are called homogeneous barycentric coordinates.
With above notation, the homogeneous barycentric coordinates of are all zero, except the one of index. When working over the real numbers, the points whose all normalized barycentric coordinates are nonnegative form the convex hull of which is the simplex that has these points as its vertices.
With above notation, a tuple such that
does not define any point, but the vector
is independent from the origin. As the direction of this vector is not changed if all are multiplied by the same scalar, the homogeneous tuple defines a direction of lines, that is a point at infinity. See below for more details.
Relationship with Cartesian or affine coordinates
Barycentric coordinates are strongly related to Cartesian coordinates and, more generally, affine coordinates. For a space of dimension, these coordinate systems are defined relative to a point, the origin, whose coordinates are zero, and points whose coordinates are zero except that of index that equals one.A point has coordinates
for such a coordinate system if and only if its normalized barycentric coordinates are
relatively to the points
The main advantage of barycentric coordinate systems is to be symmetric with respect to the defining points. They are therefore often useful for studying properties that are symmetric with respect to 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.
Relationship with projective coordinates
Homogeneous barycentric coordinates are also strongly related with some projective coordinates. 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.The projective completion of an affine space of dimension is a projective space of the same dimension that contains the affine space as the complement of a hyperplane. The projective completion is unique up to an isomorphism. The hyperplane is called the hyperplane at infinity, and its points are the points at infinity of the affine space.
Given a projective space of dimension, a projective frame is an ordered set of points that are not contained in the same hyperplane. A projective frame defines a projective coordinate system such that the coordinates of the th point of the frame are all equal, and, otherwise, all coordinates of the th point are zero, except the th one.
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. This implies that the points at infinity have their last coordinate equal to zero, and that the projective coordinates of a point of the affine space are obtained by completing its affine coordinates by one as th coordinate.
When one has points in an affine space that define a barycentric coordinate system, this is another projective frame of the projective completion that is convenient to choose. This frame consists of these points and their centroid, that is the point that has all its barycentric coordinates equal. In this case, the homogeneous barycentric coordinates of a point in the affine space are the same as the projective coordinates of this point. A point is at infinity if and only if the sum of its coordinates is zero. This point is in the direction of the vector defined at the end of.
Barycentric coordinates on triangles
In the context of a triangle, barycentric coordinates are also known as area coordinates or areal coordinates, because the coordinates of P with respect to triangle ABC are equivalent to the ratios of the areas of PBC, PCA and PAB to the area of the reference triangle ABC. Areal and trilinear coordinates are used for similar purposes in geometry.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.
Consider a triangle with vertices,, in the x,y-plane,. One may regard points in as vectors, so it makes sense to add or subtract them and multiply them by scalars.
Each triangle has a signed area or sarea, which is plus or minus its area:
The sign is plus if the path from to to then back to goes around the triangle in a counterclockwise direction. The sign is minus if the path goes around in a clockwise direction.
Let be a point in the plane, and let be its normalized barycentric coordinates with respect to the triangle, so
and
Normalized barycentric coordinates 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.
Specifically, let
is a parallelogram because its pairs of opposite sides, represented by the pairs of displacement vectors, and, are parallel and congruent.
Triangle is half of the parallelogram, so twice its signed area is equal to the signed area of the parallelogram, which is given by the determinant whose columns are the displacement vectors and :
Expanding the determinant, using its alternating and multilinear properties, one obtains
so
Similarly,
,
To obtain the ratio of these signed areas, express in the second formula in terms of its barycentric coordinates:
The barycentric coordinates are normalized so, hence . Plug that into the previous line to obtain
Therefore
.
Similar calculations prove the other two formulas
.
Trilinear coordinates of are signed distances from to the lines BC, AC, and AB, respectively. The sign of is positive if and lie on the same side of BC, negative otherwise. The signs of and are assigned similarly. Let
,,.
Then
where, as above, sarea stands for signed area. All three signs are plus if triangle ABC is positively oriented, minus otherwise. The relations between trilinear and barycentric coordinates are obtained by substituting these formulas into the above formulas that express barycentric coordinates as ratios of areas.
Switching back and forth between the barycentric coordinates and other coordinate systems makes some problems much easier to solve.