The Möller–Trumbore ray-triangle intersection algorithm, named after its inventors Tomas Möller and Ben Trumbore, is a fast method for calculating the intersection of a ray and a triangle in three dimensions without needing precomputation of the plane equation of the plane containing the triangle.
[1] Among other uses, it can be used in computer graphics to implement ray tracing computations involving triangle meshes.
[2] The ray is defined by an origin point
and a direction vector
Every point on the ray can be expressed by
, where the parameter
ranges from zero to infinity.
The triangle is defined by three vertices, named
The plane that the triangle is on, which is needed to calculate the ray-triangle intersection, is defined by a point on the plane, such as
, and a vector that is orthogonal to every point on that plane, such as the cross product between the vector from
are any points on the plane.
First, find out if the line produced by the ray intersects with the plane that the triangle is on, and if it does, find the coordinates of that intersection.
The only way that the line will not intersect the plane is if the ray's direction vector is parallel to the plane.
[3] When this happens, the dot product between the ray's direction vector and the plane's normal vector will be zero.
Otherwise, the ray does intersect the plane somewhere, but not necessarily within the triangle.
Using barycentric coordinates, any point on the triangle can be expressed as a convex combination of the triangle's vertices: The coefficients must be non-negative and sum to 1, so
is any point on the plane.
are vectors on the edge of the triangle, and together, they span a plane (which goes through the origin).
Each point on that plane can be written as
to "move" that point onto the plane that the triangle is on.
To find
for a particular intersection, set the ray expression equal to the plane expression, and put the variables on one side and the constants on the other.
This is a system of linear equations with three equations (one each for
), and can be represented as a matrix-vector multiplication.
This equation will always have a solution when the matrix has three linearly independent column vectors in
This happens if and only if the triangle vertices aren't collinear and the ray isn't parallel to the plane.
The algorithm can use Cramer's Rule to find the
values for an intersection, and if it lies within the triangle, the exact coordinates of the intersection can be found by plugging in
to the ray's equation.
The following is an implementation of the algorithm in C++: The following is an implementation of the algorithm in Rust using the glam crate: The following is an implementation of the algorithm in Java using javax.vecmath from Java 3D API: