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There are 2 observers both facing the same direction (let's assume positive y-axis in a 3D system). Initially, the observation vectors are parallel. The second observer spots the target. So, in terms of spherical co-ordinates, the observer tilts by angle $\phi$, pans by angle $\theta$ and measures the range of the target from it's position to be $R$. it Now, the first observer is only aware of the following information, the relative position of the second observer from the first observer $(\Delta x, \Delta y, \Delta z)$ and the observation information from the second observer $(R, \theta, \phi)$.

For an example, he first observer is at the origin $(0, 0, 0)$ and the second observer is at the position $(2,0,2)$ and the target is at $(1,1,1)$. Initially, the observers are both looking at the +y-axis.

The second observer tilts down by 45$\circ$ ($\phi=-\pi/4$), and pans left by 45$\circ$ ($\theta=-\pi/4$). The distance between the two points is measured ($R=\sqrt3$).

The first observer is given the following inputs:

  1. $(\Delta x, \Delta y, \Delta z)$ = $(2, 0, 2)$
  2. $(R, \theta, \phi)$ = $(\sqrt3, -\pi/4, -\pi/4)$

Based on this information alone, how can I calculate the required pan($\theta^F$) and tilt($\phi^F$) of the first observer's observation vector so that it would be looking directly at the target found by the second observer?

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  • $\begingroup$ @saulspatz I think the thing to do here is to convert the spherical coordinates to cartesian to get the relative position of the target and then use that to get spherical coordinates from the first observer. $\endgroup$ – Sam Fischer Jun 5 '18 at 0:50
  • $\begingroup$ Yes, that's right. You convert the spherical coordinates to cartesian, calculate the offset from the first observer in cartesian coordinates, and then convert that to spherical coordinates. $\endgroup$ – saulspatz Jun 5 '18 at 2:13

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