# Möller, Trumbore triangle intersection

Tomas Möller and Ben Trumbore wrote a tiny paper with the title “Fast, Minimum Storage Ray/Triangle Intersection”. It describes an algorithm that can compute the intersection of a ray with a triangle. The paper was published in 1997. It contains one sentence that should draw your attention to this paper: “…, we believe it is the fastest ray/triangle intersection routine for triangles …”. I call the paper tiny because it consists only of 7 pages. Two pages of the paper consist mainly of a C implementation of the described algorithm. One page of the paper contains only some nice pictures. The paper is very brief. According to Google Scholar, the paper was cited more than 1300 times. I suggest you to read this paper now and then come back to this page.

My implementation of the algorithm looks like this:

```
// Möller–Trumbore intersection algorithm
// Fast Minimum Storage Ray/Triangle Intersection
template <typename ScalarType>
bool intersect_triangle_moeller_trumbore(const RayType<3, ScalarType>& ray,
const Point3<ScalarType>& v0,
const Point3<ScalarType>& v1,
const Point3<ScalarType>& v2,
ScalarType& t,
ScalarType& u,
ScalarType& v) {
const ScalarType EPSILON = ScalarType{0.000001};
// find vectors for two edges sharing vert0
auto edge1 = v1 - v0;
auto edge2 = v2 - v0;
// begin calculating determinant - also used to calculate U parameter
auto pvec = ray.direction.cross(edge2);
// if determinant is near zero, ray lies in plane of triangle
auto det = edge1.dot(pvec);
if (det > -EPSILON && det < EPSILON)
return false;
auto inv_det = ScalarType{1.0} / det;
// calculate distance from vert to ray origin
auto tvec = ray.origin - v0;
// calculate U parameter and test bounds
u = tvec.dot(pvec) * inv_det;
if(u < ScalarType{0.0} || u > ScalarType{1.0}) {
return false;
}
// prepare to test V parameter
auto qvec = tvec.cross(edge1);
// calculate V parameter and test bounds
v = ray.direction.dot(qvec) * inv_det;
if(v < ScalarType{0.0} || u + v > ScalarType{1.0}) {
return false;
}
// calculate t, ray intersects triangle
t = edge2.dot(qvec) * inv_det;
return true;
}
```

I just replaced the doubles with a vector and ray type and tried to stick as close as possible to the original implementation.
Instead of using an `int`

as a return code, I switched to `bool`

.
Since I do not need support for back face culling I removed this part.

The interesting part of my implementation is my unit tests.

```
TEST(intersect_triangle_moeller_trumbore, trivial_hit) {
// Arrange
// setup ray
Point3f ray_origin{0.f,0.f,100.f};
Vector3f ray_direction{0.f,0.f,-1.f};
Ray3f ray{ray_origin, ray_direction, 0.f, 1000.f};
// counter clockwise is front facing
std::vector<Point3f> vertices = {
{-1.f, -1.f, 0.f},
{1.f, -1.f, 0.f},
{0.f, 1.f, 0.f},
};
float t = -1.f;
float u = -1.f;
float v = -1.f;
// Act
bool hit = intersect_triangle_moeller_trumbore(ray,
vertices[0],
vertices[1],
vertices[2],
t, u, v);
// Assert
EXPECT_THAT(hit, true);
EXPECT_THAT(t, 100);
EXPECT_THAT(u, 0.25);
EXPECT_THAT(v, 0.5);
}
```

Maybe you ask yourself why is $u = 0.25$ and $v = 0.5$. Take a look at the paper again. It says: $T(u,v) = (1-u-v)V_0+uV_1+uV_2$. The total area of the triangle in the above unit test is $2$. The intersection point splits the triangle into three sub-triangles. Two of them have the same size the other one has the area $1$. This gives us the barycentric coordiantes $(0.25, 0.25, 0.5)$.

Here is another interesting unit test. It is almost the same test, but this time only the ray direction is inverted. The idea here is to miss the triangle (no hit).

```
TEST(intersect_triangle_moeller_trumbore, hit_even_it_is_a_miss) {
// Arrange
// setup ray
Point3f ray_origin{0.f,0.f,100.f};
Vector3f ray_direction{0.f,0.f,1.f};
Ray3f ray{ray_origin, ray_direction, 0.f, 1000.f};
// counter clockwise is front facing
std::vector<Point3f> vertices = {
{-1.f, -1.f, 0.f},
{1.f, -1.f, 0.f},
{0.f, 1.f, 0.f},
};
float t = -1.f;
float u = -1.f;
float v = -1.f;
// Act
bool hit = intersect_triangle_moeller_trumbore(ray, vertices[0], vertices[1], vertices[2], t, u, v);
// Assert
EXPECT_THAT(hit, true);
EXPECT_THAT(t, -100);
EXPECT_THAT(u, 0.25);
EXPECT_THAT(v, 0.5);
}
```

To my surprise, the algorithm still reports a hit (returns `true`

),
but also gives a negative distance.
When just copy & pasting the implementation without recognizing this this can cause long and complicated debug sessions.

# Further reading

Möller, Trumbore triangle intersection is a good start if you want to implement your own ray triangle intersection.

During searching for ray triangle intersections I stumbled also across the paper “Yet faster ray-triangle intersection (using SSE4)”.

If you ask me which algorithm to choose for ray triangle intersection I would suggest you the paper “Watertight Ray/Triangle Intersection”.

A good CPU ray intersection library is Embree