Two point raytracing for reflection off a 3D plane |

Formally, let the ray parameter be in the open interval . Starting from the origin, Snell's law says that

gives the horizontal displacement of the ray from the origin when it reaches depth . Taking two derivatives of this formula with respect to , we have

, .

At a glance one sees that the second derivative is a quantity guaranteed to be positive in . By Thorlund-Petersen (2004), Newton's method applied to finding the for a ray that reaches a given at given depth is therefore globally convergent. (Technically, we do need to ensure that the Newton update doesn't overshoot the range .)

Two point raytracing for reflection off a 3D plane |

2012-10-29