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Ray Triangle Intersection

To perform a ray-triangle intersection test we need:

  • A ray with a known point of origin \mathbf{o}, and direction vector \mathbf{d}.
  • A triangle with three known vertex positions \mathbf{p_0}, \mathbf{v_1} and \mathbf{v_2}.
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Ray/triangle intersection using barycentric coordinates

There are several ways of computing ray-triangle intersections. The one described here uses barycentric coordinates to parameterise the triangle using two variables u and v. The point \mathbf{p} represented by the u and v is defined:

\mathbf{p}(u,v) = (1 - u - v)\mathbf{v_0} + u\mathbf{v_1} + v\mathbf{v_2}, \; \; u \geq 0, v \geq 0, u+v \leq 1

As the parametric ray equation is

\mathbf{p}(t) = \mathbf{o} + t\mathbf{d}, \; \; t \geq 0

we can derive a ray/triangle intersection by inserting this into the parametric triangle equation:

\mathbf{o} + t\mathbf{d} = (1 - u - v)\mathbf{v_0} + u\mathbf{v_1} + v\mathbf{v_2}

or as described by Möller and Trumbore (1997):

\begin{bmatrix}-\mathbf{d}, & \mathbf{v_1} - \mathbf{v_0}, & \mathbf{v_2} - \mathbf{v_0}\end{bmatrix} \begin{bmatrix}t \\ u \\ v\end{bmatrix} = \mathbf{o} - \mathbf{v_0}

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