Barycentric Coordinates Made Clear: From a UV Triangle to a 3D Triangle

This post explains, in plain language, why a simple triangular region in a 2-dimensional parameter space can describe every point of a real triangle in 3-dimensional space. Through one clean affine formula, the 2D parameters (u, v) determine points on a 3D triangle. Once this connection is seen, concepts such as interpolation, texture mapping, geometric modelling, and FEM become much easier to understand.

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Two Worlds Connected by a Map

1) Parameter World (UV-space, 2D)

Start in a flat coordinate plane labelled by two parameters, u and v. Consider the square defined by

0 ≤ u ≤ 1
0 ≤ v ≤ 1

If we cut this square along the diagonal line u + v = 1 (or equivalently v = −u + 1), we keep only the triangular region

0 ≤ u ≤ 1
0 ≤ v ≤ 1
u + v ≤ 1

This triangle has vertices (0,0), (1,0), (0,1) and is called the standard 2-simplex. Every point inside it is a convex mixture of its three corners.

2) Geometric World (3D space)

Now pick three non-collinear points in 3-dimensional space:

A, B, C ∈ ℝ³

We define the following function:

P(u, v) = C + u(A − C) + v(B − C)

Geometrically, this means: start at C, move u units towards A, and move v units towards B. Every choice of (u, v) inside our 2D triangle produces a unique point inside the 3D triangle with vertices A, B and C.

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From Plane → Parallelogram → Triangle

The Plane Through A, B, C

By expanding the expression above,

P = C + u(A − C) + v(B − C)
  = uA + vB + (1 − u − v)C

we see that the three coefficients always satisfy

u + v + (1 − u − v) = 1.

This forces P to lie on the plane through A, B, and C.

Parallelogram (UV in a Square)

If u and v are simply constrained by 0 ≤ u ≤ 1 and 0 ≤ v ≤ 1, but u + v is unrestricted, then P sweeps out the parallelogram anchored at C with edges A − C and B − C.

Triangle (Cut the Square in Half)

To restrict the points to the actual triangle ABC, we require the third weight

1 − u − v ≥ 0,

which is equivalent to

u + v ≤ 1.

Together these inequalities,

u ≥ 0,  v ≥ 0,  u + v ≤ 1,

define precisely the triangular region in UV-space that maps onto the triangle ABC in 3D.

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Barycentric Coordinates (Weights That Sum to 1)

Because

P = uA + vB + (1 − u − v)C

we may rename the weights:

α = u,   β = v,   γ = 1 − u − v.

Then

P = αA + βB + γC

with

α + β + γ = 1

and

α ≥ 0,  β ≥ 0,  γ ≥ 0.

These numbers α, β, γ are called barycentric coordinates. They describe how strongly each vertex “pulls” on the point P. If all are equal (α = β = γ = 1/3), then P is the centroid of the triangle.

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Example 1 — A Concrete Calculation

Take

A = (2, 1, 0)
B = (4, 3, 1)
C = (0, 2, 5)

Let (u, v) = (0.3, 0.4). Since 0.3 + 0.4 = 0.7 ≤ 1, this lies inside the UV-triangle. Then

P = C + u(A − C) + v(B − C)
  = (0,2,5) + 0.3(2,−1,−5) + 0.4(4,1,−4)
  = (2.2, 2.1, 1.9)

Thus P lies inside the 3D triangle ABC. In barycentric form,

α = 0.3,  β = 0.4,  γ = 0.3
P = 0.3A + 0.4B + 0.3C

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Example 2 — Visual Analogy

Imagine a flat triangular sheet of metal suspended in space at points A, B and C. A point inside the sheet can be thought of as a balance of influences from its corners. If more of the “weight” lies on A (large α), the point is pulled closer to A. If all three weights are equal, the point is exactly at the centre of mass — the centroid.

This interpretation makes clear why the coefficients must be non-negative: otherwise we leave the triangle and move beyond its physical boundary.

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Why This Clicks (and Where It’s Used)

• Universality: The same UV simplex can describe triangles in ℝ², ℝ³ or any dimension.
• Convexity: Non-negative weights that sum to 1 keep points inside the triangle.
• Affine freedom: No axes or origin are needed — only the vertices matter.
• Applications: Texture mapping, colour blending, triangle meshes, finite-element interpolation, shading, geometric modelling.

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Quick Checklist

• Plane:           P = C + u(A − C) + v(B − C)
• Parallelogram:   0 ≤ u ≤ 1,  0 ≤ v ≤ 1
• Triangle ABC:    u ≥ 0,  v ≥ 0,  u + v ≤ 1
• Weights:         α = u,  β = v,  γ = 1 − u − v
• Interior:        α, β, γ ≥ 0  and  α + β + γ = 1

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Summary

A triangle in 3D is the image of the standard 2D simplex under the affine map P(u,v) = C + u(A − C) + v(B − C). The UV inequalities u ≥ 0, v ≥ 0, u + v ≤ 1 become the barycentric conditions α, β, γ ≥ 0 with α + β + γ = 1. Hence a small triangular region in parameter space determines every point of a real geometric triangle.

This simple correspondence is the backbone of interpolation in computer graphics, numerical simulation, and more. Once recognised, it reveals a beautiful unity between algebra and geometry.