Image Depth in ADCIGs

Figure A-1 shows the basic construction to compute the image depth in ADCIGs based on the image depth in SOCIGs. Triangles $ABD$ and $CBD$ are congruent since they have one side common and the other equal because $\vert AB\vert=\vert BC\vert=h_\xi$. Therefore, $\theta=\pi/2-\beta_r+\delta$. Also, triangles AED and FCD are congruent because $\vert AD\vert=\vert CD\vert$ and also $\vert AE\vert=\vert CF\vert$ (, ). Therefore, the angle $\delta$ in triangle $DCF$ is the same as in triangle $AED$. We can compute $\delta$ from the condition

$$\begin{eqnarray*} \theta+\delta+\beta_s&=&\frac{\pi}{2},\\ \frac{\pi}{2}-\beta_... ...+\beta_s&=&\frac{\pi}{2},\\ \delta&=&\frac{\beta_r-\beta_s}{2}. \end{eqnarray*}$$

The depth of the image point in the ADCIG, from triangle ABC, is therefore

$$z_{\xi_\gamma}=z_\xi+z^*=z_\xi+(\mbox{sign}(h_\xi))h_\xi\cot\left(\frac{\pi}{2}-\beta_r+\delta\right).$$ (5)

Replacing the expression for $\delta$ we get, after some simplification (and taking sign$(h_\xi)=-1$)

$$z_{\xi_\gamma}=z_\xi+z^*=z_\xi-h_\xi\tan\left(\frac{\beta_r+\beta_s}{2}\right)=z_\xi-h_\xi\tan(\gamma).$$ (6)

mul-sktch17 Figure 1. Sketch to show the computation of the image depth in an ADCIG.