Definition:Barycentric Coordinates

Definition

Barycentric coordinates are a set of numbers representing the position of a point in a Euclidean space with respect to a set of fixed points.

In $n$ dimensional space a total of $n + 1$ points must be used.

$3$ Dimensions

In $3$ dimensional space, barycentric coordinates are a set of $4$ numbers representing the position of a point as follows:

Let $p_0, p_1, p_2, p_3$ be fixed non-coplanar points, such that $p_i = \tuple {x_i, y_1, z_i}$.

Then an arbitrary point $p$ can be expressed in the form:

$p = \lambda_0 p_0 + \lambda_1 p_1 + \lambda_2 p_2 + \lambda_3 p_3$

such that:

$\lambda_0 + \lambda_1 + \lambda_2 + \lambda_3 = 0$

The set $\set {\lambda_0, \lambda_1, \lambda_2, \lambda_3}$ consists of the barycentric coordinates of $p$.

Also see

• Results about barycentric coordinates can be found here.

Sources

• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): barycentric coordinates
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): barycentric coordinates