raytriangle

raytriangle intersection is a fundamental operation in computer graphics and computational geometry. It determines whether a ray, an infinitely extending line with a defined origin and direction, intersects with a triangle, a planar polygon defined by three vertices. This intersection test is crucial for tasks such as ray tracing, collision detection, and picking objects in 3D environments.

The core of the ray-triangle intersection algorithm involves several steps. First, the ray is mathematically represented as $R(t) = O + tD$, where $O$ is the origin point, $D$ is the direction vector, and $t$ is a scalar parameter representing distance along the ray. The triangle is defined by its vertices $V_0$, $V_1$, and $V_2$.

A common approach to solve this is to first determine if the ray intersects the plane containing the triangle. If it does, a parameter $t$ is calculated. Then, the intersection point on the plane is projected onto the triangle. This projection is typically done using barycentric coordinates. If the barycentric coordinates of the intersection point lie within the bounds of the triangle (i.e., they are all non-negative and sum to 1), then an intersection occurs. If the intersection point lies outside the triangle, or if the ray is parallel to the plane, no intersection with the triangle is registered.

Various optimizations exist for the ray-triangle intersection test, such as the Möller-Trumbore algorithm, which efficiently calculates the intersection point and barycentric coordinates simultaneously, avoiding intermediate plane-intersection calculations. These algorithms are highly optimized for performance due to their frequent use in real-time rendering applications.