Barycentric Coordinates — Definition, Formula & Examples

Barycentric coordinates describe the location of a point relative to a triangle by assigning a weight to each vertex. A point's position is the weighted average of the three vertices, where the weights sum to 1.

Given a triangle with vertices $A$ , $B$ , $C$ , the barycentric coordinates of a point $P$ are scalars $(\lambda_1, \lambda_2, \lambda_3)$ such that $P = \lambda_1 A + \lambda_2 B + \lambda_3 C$ and $\lambda_1 + \lambda_2 + \lambda_3 = 1$ . When all three coordinates are non-negative, $P$ lies inside or on the boundary of the triangle.

Key Formula

P = \lambda_1 A + \lambda_2 B + \lambda_3 C, \quad \lambda_1 + \lambda_2 + \lambda_3 = 1

Where:

$P$ = The point being described
$A, B, C$ = The three vertices of the reference triangle
$\lambda_1, \lambda_2, \lambda_3$ = Weights (barycentric coordinates) assigned to each vertex

How It Works

Each barycentric coordinate represents the ratio of the sub-triangle area opposite that vertex to the total triangle area. For instance,

\lambda_1

equals the area of triangle

PBC

divided by the area of triangle

ABC

. If

P

sits exactly at vertex

A

, its coordinates are

(1, 0, 0)

. The centroid of the triangle always has coordinates

(\tfrac{1}{3}, \tfrac{1}{3}, \tfrac{1}{3})

. A negative coordinate means the point lies outside the triangle, on the far side of the opposite edge.

Worked Example

Problem: A triangle has vertices

A = (0, 0)

B = (6, 0)

C = (0, 4)

. Find the Cartesian position of the point with barycentric coordinates

(0.5,\, 0.25,\, 0.25)

Step 1: Apply the formula

P = \lambda_1 A + \lambda_2 B + \lambda_3 C

using the given weights.

P = 0.5(0,0) + 0.25(6,0) + 0.25(0,4)

Step 2: Compute each weighted term and add.

P = (0,0) + (1.5, 0) + (0, 1) = (1.5,\, 1)

Step 3: Verify the weights sum to 1:

0.5 + 0.25 + 0.25 = 1

. The point lies inside the triangle since all coordinates are positive.

Answer: The point is at Cartesian coordinates

(1.5,\, 1)

Why It Matters

Barycentric coordinates are essential in computer graphics for texture mapping, ray-triangle intersection tests, and interpolating color or depth across a triangle. In physics, they locate a system's center of mass. They also simplify many classic triangle geometry proofs by replacing distance-based reasoning with algebraic ratios.

Common Mistakes

Mistake: Forgetting that the three coordinates must sum to 1 for normalized barycentric coordinates.

Correction: Always check

\lambda_1 + \lambda_2 + \lambda_3 = 1

. If you compute unnormalized weights (e.g., sub-triangle areas), divide each by the total triangle area to normalize.