Computing Barycentric Coordinates

Chris Tralie

In the GUI below, choose three points in the plane the plane (z = 0) to make a triangle, and then select a fourth point. Left click to select the points, and right click to delete the most recently selected point. Once the fourth point is selected, that triangles will be drawn. The barycentric coordinates can then be computed by the area ratios shown below

NOTE: This GUI runs in the plane for simplicity (all z coordinates are zero), but the same phenomenon will happen in 3D.

Area Ratios

The ratios of the areas of triangles can be used to compute the barycentric coordinates. If the point d is inside the triangle abc, then the following rules apply:

\[ \alpha = \text{area} \frac{\bigtriangleup bcd}{\bigtriangleup abc}, \beta = \text{area} \frac{\bigtriangleup acd}{\bigtriangleup abc}, \gamma = \text{area} \frac{\bigtriangleup abd}{\bigtriangleup abc} \]

Notice how in this case, alpha + beta + gamma = 1

If the point is outside the triangle (either in the same plane but outside, or above or below the plane of the triangle), then

\[ \text{area} ({\bigtriangleup abc}) < \text{area} ({\bigtriangleup abd}) + \text{area} ({\bigtriangleup acd}) + \text{area} ({\bigtriangleup bcd}) \]