Barycentric coordinate system

Barycentric coordinates ( also homogeneous barycentric coordinates ) are used in linear algebra and geometry to describe the location of points with respect to a given distance, a given triangle, a given tetrahedron, or more generally a given simplex. , The point represented by the coefficients of a Affinkombination ( ie, a linear combination of points, in which the sum of coefficients is 1).

They are a special case of homogeneous coordinates.

• 3.1 Calculation
• 3.2 Features
• 3.3 Conversion Formulas

Examples

Points of a line

Given a segment [ AB ]. For each point of the line AB is barycentric coordinates can specify where they have not been clearly defined. In order to achieve uniqueness, are often used normalized barycentric coordinates, that is, the barycentric coordinates, the sum of which is equal to 1. The values ​​given in the sketch for some sample points barycentric coordinates are normalized. When you remove the scaling, so you could, for example, express the point A by (5.0 ) or the midpoint of [AB ] by ( 1.1 ). In general, the multiplication of all coordinates with the same real number ( not 0) leads again to barycentric coordinates.

Points of a plane

In a plane, the barycentric coordinates are related to a given triangle. Entered in the drawing coordinates are normalized.

General definition

X1, ..., xn are the vertices of the simplex in the vector space A. If the following equation is satisfied for a point P from O,

We call the coefficients (a1, ..., an) of barycentric coordinates P x1, ..., xn. The vertices have coordinates (1, 0, 0, ..., 0 ), ( 0, 1, 0, ..., 0 ), ..., (0, 0, 0, ..., 1 ). Barycentric coordinates are not unique: For each non-zero b ( b a1, ..., b in ) also barycentric coordinate of p.

If the coordinates are positive, the point p is in the convex hull of x1, ..., xn, so the simplex with these vertices. The representation of a point inside a convex hull as the sum of vertices of a simplex is called affine combination or barycentric combination.

Let us masses in the ratio of a1, ..., an at the vertices of the simplex before, so is the center of mass ( the barycenter ) in p. This is the origin of the term barycentrically, which was introduced in 1827 by August Ferdinand Möbius.

Barycentric coordinates in the triangle geometry

In addition the triangular geometry trilinear coordinates barycentric coordinates are frequently used to describe the positions of marked points. In addition to the triple notation and the notation is in use.

Calculation

The barycentric coordinates of a point are given by

This mean and oriented triangular faces, ie the surfaces of said triangles obtained with a positive sense of rotation ( counterclockwise ) a positive sign, otherwise a negative.

Properties

Three points with barycentric coordinates, and then lie exactly on a straight line when

Applies. This relationship also allows the preparation of linear equations for barycentric coordinates. Such equations have the form

Wherein at least one of the real coefficient, and 0 must be different.

Three straight with the barycentric equations, and intersect if and only at one point or are parallel, if the condition

Is satisfied.

The parallelism of two straight lines with equations and can be checked with the criterion

Conversion formulas

There is a simple relationship between trilinear and barycentric coordinates: Does the given point, the trilinear coordinates so barycentric coordinates of that point. And thereby denote the side lengths of the triangle.

The conversion to Cartesian coordinates is done by determination of the weighted average. Are barycentric co-ordinates of a point and the Cartesian coordinates of the corners of the given triangle, then:

Generalized barycentric coordinates

Barycentric coordinates ( a1, ..., an), which are defined with respect to a polyhedron instead of with respect to a simplex, generalized barycentric coordinates are known. This is further required that the equation

Is satisfied, where x1, ..., xn are the vertices of the given polytope here. The definition is thus formally unchanged, but you have a simplex with n vertices in a vector space of dimension at least n-1 be included while polytopes can also be embedded in subspaces of lower dimension. The simplest example is a square in the plane. As a consequence, even the normalized barycentric coordinates for a generalized polytope generally are not unambiguously determined, although this is the case for normalized barycentric coordinates with respect to a simplex.

Generalized barycentric coordinates are used especially in computer graphics and geometric modeling. Where three-dimensional objects can often be approximated by polyhedrons, so that the generalized barycentric coordinates have a geometric meaning and to facilitate the further processing of these objects.

Barycentric interpolation

In barycentric coordinates based an interpolation method that generalizes the linear interpolation for functions of several variables.

In the case of a function of two variables ( and ) are for three points, and given the function values ​​. Please adhere to, and do not lie on a straight line. So you need to span a triangle. Is now an arbitrary point is given, we define

The normalized barycentric coordinates of are. This interpolation also works for points outside the triangle.