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![]() | Two point raytracing for reflection off a 3D plane | ![]() |
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for some scalar
By our definition of
which, using the identity,
gives the relation for
which, combined with
produces a fourth order equation for
The fourth order equation can be solved directly using algebraic formulas. Lanczos (1956) provides a clean, efficient numerical approximation, reproduced in Appendix A, that is about 10 times faster than using a general purpose numerical root finder. (Appendix B shows how to make it free of floating point divisions.)
An interesting alternative to direct solution is to apply Newton iterations to the shooting method wherein source ray parameters are repeatedly adjusted to return very near to the target receiver. This approach applies to multiple layers and multiple reflections, not just a single interface. In Appendix C, I demonstrate global convergence of that method when applied to forward ray tracing through a stack of horizontal layers.
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![]() | Two point raytracing for reflection off a 3D plane | ![]() |
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