Barycentric Coordinates

Barycentric coordinates express any point p in the plane as a unique affine combination of three non-collinear triangle vertices v₁, v₂, and v₃, directly encoding whether p lies strictly inside, on the boundary, or outside the triangle.

Affine Combinations and Signed Sub-Triangle Areas

For triangle vertices v₁, v₂, v₃ in 2D, any point p can be written as p = λ₁v₁ + λ₂v₂ + λ₃v₃ subject to the affine normalization constraint λ₁ + λ₂ + λ₃ = 1. By Cramer's rule, each weight λ_i equals the ratio of the signed area of the sub-triangle opposite v_i to the signed area of the full triangle (v₁, v₂, v₃).

Linear System and Point Classification

Solving the 3x3 linear system with rows [v_1,x, v_2,x, v_3,x], [v_1,y, v_2,y, v_3,y], and [1, 1, 1] requires a nonzero signed-area determinant det ≠ 0. Because each λ_i divides its sub-determinant by det, reversing vertex orientation from counter-clockwise to clockwise flips the sign of both numerator and denominator, leaving (λ₁, λ₂, λ₃) invariant. Point p is classified as: Inside if all λ_i > 0; Boundary if all λ_i ≥ 0 and at least one λ_i = 0; and Outside if any λ_i < 0.

Worked Example

Find the barycentric coordinates of p = (0, 1) with respect to v₁ = (-1, 0), v₂ = (2, 0), and v₃ = (0, 2):

Set up the equations: x-coordinate: -λ₁ + 2λ₂ + 0λ₃ = 0; y-coordinate: 0λ₁ + 0λ₂ + 2λ₃ = 1; affine sum: λ₁ + λ₂ + λ₃ = 1.
Solve for λ₃: From 2λ₃ = 1, we obtain λ₃ = 1/2 = 0.5.
Solve for λ₁ and λ₂: Since λ₁ = 2λ₂ and λ₁ + λ₂ = 1/2, we get 3λ₂ = 1/2 → λ₂ = 1/6 and λ₁ = 1/3.
Classify: All three weights (1/3, 1/6, 1/2) are strictly positive and sum to 1, so p lies Inside the triangle.

Experiment

Explore interior interpolation, edge boundaries, exterior extrapolation, and collinear degeneracy in https://riano.app/module/barycentric-coordinates:

Keep the default triangle (0,0), (3,0), (0,3) with p = (1,1) to verify equal weights (1/3, 1/3, 1/3) and classification Inside.
Move Point x to 0 and Point y to 0 so p coincides with v₁, yielding weights [1, 0, 0] and classification Boundary.
Set collinear vertices (-1,-1), (0,0), (1,1) to verify that the singular system reports Degenerate and hides undefined weights as Unavailable.

Limitations

Barycentric coordinates are uniquely defined only when the simplex vertices are affinely independent (non-collinear in 2D). When vertices are collinear or repeated, the system matrix is singular and no unique affine coordinate representation exists.

Common Pitfalls

Using unsigned triangle areas instead of signed determinants fails to detect points outside the triangle (where at least one signed coordinate must be negative) and breaks when vertices are listed in clockwise order.

References

Affine simplex transforms and barycentric coordinate computation follow standard computational geometry references:

SciPy Spatial Reference: scipy.spatial.Delaunay.transform (Affine Barycentric Coordinates)

Barycentric Coordinates