Barycentric Coordinates

A Barycentric Coordinate system is a coordinate system in which the location of a point is specified as a weighted sum of other masses.

In the case of a triangle, we can use the vertices of a triangle as the mass points of a barycentric coordinate system. Using the barycentric coordinate system, we can specify any point inside this triangle as a way to add some of its vertices.

The weight values used to specify the location of a point in this triangle are known as barycentric coordinate . If a point is inside a triangle, these barycentric coordinate will add up to one.

Let’s try to find how to calculate the barycentric coordinates of this point P within this triangle formed by the vertices v0, v1 and v2. This is calculated through comparing the ratio of triangles formed by this point P and the points v0, v1 and v2. So the first thing we’re going to do is take two vectors along the edges of the triangle. Like the aove image let’s take one of the example. We’ll take the vectors formed from v0 to v1 and v0 to v2 and we’ll take the cross product of those two vectors and that gives us a vector perpendicular to the triangle.

Now it turns out that the the length of this perpendicular vector is equal to the area of the parallelogram formed by doubling this triangle. So now if we divide this triangle up into three sub triangles formed with the point P and the other vertices of the triangle, we get something that looks like the first image. So if we then compare the ratio of these smaller triangles to the area of this containing triangle, that will give us the barycentric coordinate of each point.

Here is simple script to find the position of P in a triangle:

x, y, z = get_barycentric_coordinates(v0, v1, v2, P)
 
def get_barycentric_coordinates(v0, v1, v2, P):
    normal = (v1 - v0) ^ (v2 - v0)
    area = normal.length()
         
    # Compute v0
    v2_point = v2 - P
    areaPoint_v1_v2 = ((v1 - P) ^ v2_point).length()
    a = areaPoint_v1_v2 / area
     
    # Compute v1
    areaPoint_v2_v0 = (v2_point ^ (v0 - P)).length()
    b = areaPoint_v2_v0 / area    
     
    # Compute v2
    c = 1.0 - v0_position - v1_position
     
    return a, b, c

I had written another post that demostrates the real world usage of barycentric coordinates. Have a look at this and it will give you a bit more of an idea where you could think of using barycentric coordinates.

Copy Weights between Asymmetrical Meshes Using Barycentric Coordinates (Maya API)

Thank you to many people who helped me along my way to understand 3D Math myself. Won’t be here if they did not help me learn and share their knowledge. Resources and images were either done by me or found online from Wikipedia and other related websites.

Until next time.

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