Two point raytracing for reflection off a 3D plane

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## Globally convergent Newton's method for ray shooting

Quite some time ago, Bob Keyes at Mobil mentioned that Newton's method applied to shooting rays to solve the two-point problem in horizontally layered media is globally convergent, assuming, of course, that there is a solution. Specifically, there must be a solution if an initial guess at the ray parameter overshoots the target.

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

Next: APPENDIX D Up: APPENDIX C Previous: APPENDIX C

2012-10-29