Two point raytracing for reflection off a 3D plane

Introduction

For SEP-147, I calculated the response of various classic seismic algorithms on a reflection off of a plane in 3D. After wrestling with spatial geometry in old textbooks, I derived the following result from scratch using elegant, coordinate-free vector notation.

planerayfig Figure 1. Diagram of planar reflector and the points and vectors I use for calculating the reflected raypath.

Given a source location $S$ , a receiver location $R$ , and a plane ${\bf n} \cdot ( P - P_0 ) = 0$ , where ${\bf n}$ is a unit normal, to find the reflection point $P$ , drop a perpendicular ${\bf w}$ from ${\bf n}$ to the line connecting $P$ to $R$ . Snell's Law says that running ${\bf w}$ in the other direction connects to the line between $P$ and $S$ . So for some scalars $\alpha$ and $\beta$ we have

$$\begin{array}{ccc} ( R - P ) & = & \alpha ( {\bf n} + {\bf w... ...bf w} & = & 0 \\ {\bf n} \cdot ( P - P_0 ) &=& 0 \end{array}.$$

Dotting ${\bf n}$ onto the first two equations gives

$$\begin{array}{ccccc} {\bf n} \cdot ( R - P ) & = & {\bf n} \... ...- P ) & = & {\bf n} \cdot ( S - P_0 ) & = & \beta \end{array},$$

and subtracting the first two equations produces

$$( R - S ) = ( \alpha - \beta ) {\bf n} + ( \alpha + \beta ) {\bf w} ,$$

which can be solved directly for ${\bf w}$ now that we have $\alpha$ and $\beta$ . Given this ${\bf w}$ , the first equation immediately yields

$$P = R - \alpha ( {\bf n} + {\bf w} ) ,$$

the desired reflection point. This can also be described in terms of the midpoint $M$ of the source and receiver as

$$P = M - {\scriptstyle \frac{1}{2}} ( \alpha + \beta ) {\bf n } - { \scriptstyle \frac{1}{2} } ( \alpha - \beta ) { \bf w }$$    .

Two point raytracing for reflection off a 3D plane