Barycentric Coordinates

Definition

Barycentric coordinates $(\alpha, \beta, \gamma)$ express a point inside a triangle as a weighted combination of the vertices:

$\mathbf{P} = \alpha\mathbf{A} + \beta\mathbf{B} + \gamma\mathbf{C}$

where $\alpha + \beta + \gamma = 1$ and $\alpha, \beta, \gamma \geq 0$ for points inside the triangle.

Computing Barycentric Coordinates

Given triangle vertices $(x_A, y_A)$, $(x_B, y_B)$, $(x_C, y_C)$ and point $(x, y)$:

Define implicit line equation: $f_{AB}(x, y) = (y_A - y_B)x + (x_B - x_A)y + x_A y_B - x_B y_A$

Then: $\alpha = \frac{f_{BC}(x, y)}{f_{BC}(x_A, y_A)}$ $\beta = \frac{f_{AC}(x, y)}{f_{AC}(x_B, y_B)}$ $\gamma = 1 - \alpha - \beta$

Interpolating Attributes

Any attribute $Q$ (colour, normal, texture coordinates) can be interpolated:

$Q(x, y) = \alpha Q_A + \beta Q_B + \gamma Q_C$

Used for:

• Colour interpolation (Gouraud shading)
• Normal interpolation (Phong shading)
• Texture coordinate interpolation

Point-in-Triangle Test

A point is inside the triangle if and only if: $\alpha > 0 \text{ and } \beta > 0 \text{ and } \gamma > 0$

Summary

• Barycentric coordinates express points as vertex weights
• Sum to 1 inside triangle; all positive for interior points
• Enable smooth attribute interpolation across triangle

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Past Paper Questions

2025 Paper 3 Question 4(b)(iv): What are barycentric coordinates, and where are they needed in the OpenGL rendering pipeline? [3 marks]