Calculating the first triangle hit by the viewing ray after it leaves the image pixel (Figure 7.1) is straightforward if we neglect the computational performance. Starting with the triangle coordinates, focal point, and the ray direction (vector), the closed-form solution involves basic operations from analytic geometry, including dot products, cross products, and the plane equation [320]. For each triangle, it must be determined whether the ray intersects it. If not, then the next triangle is considered. If it does, then the intersection is recorded as the candidate solution only if it is closer than the closest intersection encountered so far. After all triangles have been considered, the closest intersection point will be found. Although this is simple, it is far more efficient to arrange the triangles into a 3D data structure. Such structures are usually hierarchical so that many triangles can be eliminated from consideration by quick coordinate tests. Popular examples include BSP-trees and Bounding Volume Hierarchies [42,85]. Algorithms that sort geometric information to obtain greater efficiently generally fall under *computational geometry* [54]. In addition to eliminating many triangles from quick tests, many methods of calculating the ray-triangle intersection have been developed to reduce the number of operations. One of the most popular is the *Möller-Trumbore intersection algorithm* [217].

Steven M LaValle 2016-12-31