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You are given an arbitrary angle $h$. It can be any angle from $0^\circ$ to $360^\circ$. You are also given an angle $v$, which is guaranteed to be between $-90^\circ$ and $90^\circ$. Find angle $a$ given these two angles. What is the equation to solve for any combination of these? I am using this to project 3D vectors into a single 2D isometric angle (for a video game that is 2D isometric using 3D physics).

Here is a truth table: If $h=0^\circ, a=v$.
If $h=90^\circ, a=90^\circ$ if $v \geq 0^\circ$, $-90^\circ$ otherwise.
If $h=180^\circ, a=180^\circ - v$.
If $h=270^\circ, a=270^\circ$. (always!)

Here is an illustration:

demonstration

N. F. Taussig
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Anixias
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    This tutorial explains how to typeset mathematics on this site. – N. F. Taussig Apr 23 '18 at 22:47
  • I'm not seeing the pattern... – ericw31415 Apr 23 '18 at 23:00
  • The pattern is that the angle h sort of controls the range of the answer, and angle v gives the exact answer within that range. For h = 90 or h = 270, the range is limited to one or two possible answers. I tried playing around with manipulating the cosines and sines and then using arccos and arcsin to get the angle out of them, but I was unsuccessful. – Anixias Apr 24 '18 at 00:26

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