Barycentric on Triangle - Computer Graphics Step by Step

Use exact barycentric weights to decide whether a point is inside a triangle.

barycentric-on-triangle Barycentric weights describe a point as a weighted mix of the triangle vertices. If the weights sum to one and stay nonnegative, the point is inside or on the triangle.

Example

Use barycentric weights to decide whether a point is inside a triangle. This is exact integer rasterization; real rendering anti-aliases these edges.

highlighted = computed this step

Step 1 — Set up

Set up the triangle and test point.

A=(1,1),\ B=(6,1),\ C=(2,5),\ P=\hlmath{(3,2)}

Step 2 — Compute alpha

Compute alpha from the area ratios.

\alpha=\frac{2}{5}

Step 3 — Compute beta

Compute beta from the area ratios.

\beta=\frac{7}{20}

Step 4 — Compute gamma

Compute gamma from the area ratios.

\gamma=\frac{1}{4}

Step 5 — Check the sum

Check that the three weights sum to one.

\alpha+\beta+\gamma=\frac{2}{5}+\frac{7}{20}+\frac{1}{4}=1

Step 6 — Result

State the inside or outside verdict.

\text{inside}