Barycentric Coordinates

2 min read•1 page

Sub-Triangle Area Ratio:

1. Coordinate u: Ratio of area(PBC) to total area(ABC). 2. Coordinate v: Ratio of area(PCA) to total area(ABC). 3. Coordinate w: Ratio of area(PAB) to total area(ABC).

python

1def compute_barycentric(p, a, b, c):
2    v0, v1, v2 = b - a, c - a, p - a
3    d00 = dot(v0, v0)
4    d01 = dot(v0, v1)
5    d11 = dot(v1, v1)
6    d20 = dot(v2, v0)
7    d21 = dot(v2, v1)
8    
9    denom = d00 * d11 - d01 * d01
10    v = (d11 * d20 - d01 * d21) / denom
11    w = (d00 * d21 - d01 * d20) / denom
12    u = 1.0 - v - w
13    return u, v, w

Barycentric coordinates represent any point P within a triangle ABC as a weighted sum of its vertices: P = u A + v B + w C, where u + v + w = 1.

1 min read•1 page

Attribute blending:

1. Partition of Unity: Because u + v + w = 1, any convex combination is guaranteed to lie within the bounds of the vertex attributes. 2. Vertex Blending: Color at P is u · Color(A) + v · Color(B) + w · Color(C).

python

1# Interpolating vertex attributes (e.g. colors, normals) using weights
2def interpolate_attributes(u, v, w, attr_a, attr_b, attr_c):
3    # Linear combination
4    return u * attr_a + v * attr_b + w * attr_c

Barycentric coordinates are the basis of fragment shading. By treating u, v, and w as weights, we linearly interpolate attributes (colors, UV coordinates, normals) across the triangle face.

Point P Position X

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