Barycentric Coordinates for Point-in-Triangle Tests

Expressing Any Point as a Weighted Blend

Given triangle vertices A, B, and C, any point P in the plane can be written as:

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The triple (u, v, w) is the barycentric coordinate of P relative to the triangle. Geometric meaning:

• All coordinates positive: P is strictly inside
• One coordinate zero: P lies on an edge
• Any coordinate negative: P is outside

Edge Functions Compute Coordinates

Barycentric coordinates come from edge functions, a 2D cross product:

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For a counter-clockwise triangle ABC:

• u = edge(P, B, C) / edge(A, B, C)
• v = edge(P, C, A) / edge(A, B, C)
• w = edge(P, A, B) / edge(A, B, C)

For example, if all three numerators are non-negative for pixel center P, that pixel is inside the triangle. The denominator is twice the signed triangle area. GPUs compute these values in fixed-function rasterizer hardware for attribute interpolation.

For This Exercise

You will compute barycentric coordinates for test points and determine whether each lies inside a given triangle. This task asks you to implement edge functions and print inside/outside results. The same math underlies Pixar's REYES renderer and every ray-triangle intersection test in path tracers like PBRT.