What Are Barycentric Coordinates?

Barycentric coordinates provide a way to describe any point inside a triangle using the triangle’s own vertices as a reference. Instead of relying on the usual x–y axes, we express a point as a weighted combination of the three corners.

The Core Idea

Let a triangle have vertices A, B and C. Any point P inside (or on) the triangle can be written as

P = αA + βB + γC

The numbers α, β and γ are the barycentric coordinates of P. They indicate how strongly each vertex contributes to P.

For this expression to make geometric sense, the three weights must satisfy

α + β + γ = 1

This condition ensures that P behaves like a weighted average — a “blend” of A, B and C — rather than drifting away from the triangle. As long as all three values are non-negative (α, β, γ ≥ 0), the point lies somewhere within the triangle.

Examples

• α = 1, β = γ = 0  →  P = A
• α = β = 0.5, γ = 0  →  midpoint of AB
• α = β = γ = 1/3  →  the triangle’s centroid (its centre)

By adjusting the three weights, we move P around the interior. If one weight becomes zero, P lies on the opposite edge; if two weights are zero, P is located exactly at a vertex. Only when a weight drops below zero does P leave the triangle.

Why They Are So Useful

Barycentric coordinates are tied directly to the triangle itself. If the triangle moves, stretches, or tilts anywhere in space, the same triplet (α, β, γ) still identifies the same relative position within that new triangle. There is no dependency on the global coordinate axes or an external origin.

This makes barycentric coordinates extremely valuable in areas such as computer graphics, interpolation, mesh deformation, animation, and finite-element methods. Anytime a problem is expressed in terms of triangles, barycentric coordinates provide a natural and elegant way to describe positions.

Summary

Barycentric coordinates describe a point inside a triangle by indicating how much it leans toward each of the three vertices. They are three weights that:

• sum to 1,
• are non-negative for points inside the triangle, and
• form a weighted average of A, B and C.

P = αA + βB + γC

They offer a simple, intuitive, and highly flexible way to work with points inside a triangle — perfect for both geometric reasoning and practical computation.