On-Line Computer Graphics Notes

## Barycentric Coordinates

If we are given a frame in three-dimensional space we know how to define a local coordinate system with respect to the frame. However, given a set of points in three-dimensional space, we can also define a local coordinate system with respect to the these points. These coordinate systems are called barycentric coordinates and are discussed in these notes.

For a postscript version of these notes look here.

What are Barycentric Coordinates?

Consider a set of points , , ..., and consider the set of all affine combinations taken from these points. That is all points that can be written as

for some

This set of points forms an affine space, and the coordinates

are called the barycentric coordinates of the points of the space.

These coordinates system are frequently quite useful, and the interested student will notice that they are used extensively in working with triangles. In many cases (e.g. on a line, as shown below), this barycentric parameterization is exactly the parameterization that we usually use.

Example -- Point on a Line Segment

To give a simple example of barycentric coordinates, consider two points and in the plane. If and are scalars such that , then the point defined by

is a point on the line that passes through and . If then the point is on the line segment joining and . The following figure shows an example of a line and three points , and . These points were generated using the following s:

• : ,
• : ,
• : ,

Example -- Point in a Triangle

To give a slightly more complex example of barycentric coordinates, consider three points , , in the plane. If , , are scalars such that , then the point defined by

is a point on the plane of the triangle formed by , , . The point is within the triangle if . If any of the 's is less than zero or greater than one, the point is outside the triangle. If any of the 's is zero, we reduce to the example above and note that is on one of the lines joining the vertices of the triangle. The following figure shows an example of such a triangle and three points , and , these points were calculated using the following 's:

• : , .
• : , , .
• : , , .

Frames and Barycentric Coordinates

There is a natural way to convert the local coordinates of a frame to barycentric coordinates for a certain set of points. Suppose we are given a frame for an affine space . Then we can write any point in the space uniquely as

where are the local coordinates of the point with respect to the frame . If we define the points by

(i.e., the origin of the frame and the points obtained by adding the coordinate vectors to the origin) and define to be

then we can see that can be written as

or equivalently, in an affine way as,

where

In this form, the values are barycentric coordinates of relative to the points

How can Vectors be Represented?

Following the above methods, we can also express the vectors of an affine space in terms of the points. In this case, if we are given the frame then for any vector , we can write as

for some constants (since the vectors of the frame are assumed to be linear independent). Now, if we define

If we define the points by

then

or equivalently, in an affine way as,

where now we have that .

## No References!

Summary

Barycentric coordinates are another important method of introducing coordinates into an affine space. If the coordinates sum to one, they represent a point ; if the coordinates sum to zero, they represent a vector.