triangleIntersect: Ray-Triangle Intersection Algorithm

Möller-Trumbore Method for Efficient Ray Testing

The triangle intersection function implements the Möller-Trumbore algorithm for computing ray-triangle intersections in 3D space. This method efficiently determines both intersection existence and parametric distance along the ray.

Given triangle vertices $A$, $B$, $C$ and ray with origin $O$ and direction $\vec{d}$, the algorithm computes edge vectors and cross products:

$$\begin{align} \vec{v_1} &= B - A \\ \vec{v_2} &= C - A \\ \vec{h} &= \vec{d} \times \vec{v_2} \\ \end{align}$$

The determinant $a = \vec{v_1} \cdot \vec{h}$ determines ray-plane parallelism. When $|a|$ approaches zero, the ray lies parallel to the triangle plane.

Barycentric Coordinate Calculation

The algorithm computes barycentric coordinates $(u, v)$ within the triangle:

$$\begin{align} \vec{s} &= O - A \\ u &= \frac{\vec{s} \cdot \vec{h}}{a} \\ \vec{q} &= \vec{s} \times \vec{v_1} \\ v &= \frac{\vec{d} \cdot \vec{q}}{a} \end{align}$$

Valid intersections satisfy $u \geq 0$, $v \geq 0$, and $u + v \leq 1$.

Distance Parameter

The parametric distance $t$ along the ray is calculated as:

$t = \frac{\vec{v_2} \cdot \vec{q}}{a}$

When $t < 0$, the intersection occurs behind the ray origin. The function returns $t$ for valid intersections or a large value ($9999999.9$) for misses.

ライブエディター

const fragment = () => Scope(() => {
      const tri = Triangle({
              a: vec3(-0.5, -0.4, 0),
              b: vec3(0.5, -0.4, 0),
              c: vec3(0, 0.6, 0)
      })
      const rayOrigin = vec3(uv.mul(2).sub(1), -1)
      const rayDir = vec3(0, 0, 1)
      const dist = triangleIntersect(tri, rayOrigin, rayDir)
      const hit = dist.step(1000)
      const depth = hit.mul(0.8).add(0.2)
      return vec4(vec3(depth), 1)
})