I have a flashlight angled at [math]\phi[/math] from zenith (the z-axis).

the flashlight can be rotated around the z-axis so the beam forms a cone (angled at [math]\phi[/math] from zenith).

 

Moreover, the bulb in the flashlight can also be angled [math]\phi[/math], so the resulting angle from zenith can vary from [math]0[/math] to [math]2\phi[/math] degrees.

 

The question is, how do I transform a cartesian coordinate representation in space, into the two axes? So, for instance, [math](x,y,z)[/math] becomes [math](r,\theta_1,\theta_2)[/math] where [math]\theta_1[/math] is the angle of the flashlight itself. and [math]\theta_2[/math] the angle of the bulb. [math]r[/math] is the radial distance which is probably [math]\sqrt{x^2+y^2+z^2}[/math].

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I just realized that there are several solutions to the cartesian coordinates. Perhaps it is better to define [math]\theta_2[/math] as the difference between the bulb and the flashlight. Thus [math]\theta_2 = 0[/math] when the resulting beam angle from zenith is [math]2\phi[/math].

In that case, what would the transform then look like?

No-one an expert on coordinate transforms?

You're looking for something like spherical polar co-ordinates.

Yes I know. However, it's with a twist because the actual position is a combination of the angles.