Computation of Image Depth in ADCIGs

 

mul_sktch17 Figure 20 Sketch to show the computation of the image depth in an ADCIG.

Figure  shows the basic construction to compute the image depth in ADCIGs based on the image depth is SOCIGs. Triangles ABD and CBD are congruent since they have one side common and the other equal because |AB|=|BC|=$h_\xi$. Therefore, $\theta=\pi/2-\beta_r+\delta$. Also, triangles AED and FCD are congruent because |AD|=|CD| and also |AE|=|CF| Biondi and Symes (2004). Therefore, the angle $\delta$ in triangle DCF is the same as in triangle AED. We can compute $\delta$ from the condition The depth of the image point in the ADCIG, from triangle ABC, is therefore

$$z_{\xi_\gamma}=z_\xi+z^*=z_\xi+(\mbox{sign}(h_\xi))h_\xi\cot\left(\frac{\pi}{2}-\beta_r+\delta\right).$$ (55)

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

$$z_{\xi_\gamma}=z_\xi+z^*=z_\xi-h_\xi\tan\left(\frac{\beta_r+\beta_s}{2}\right)=z_\xi-h_\xi\tan(\gamma).$$ (56)

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