Background:
I'm trying to build a lightweight antenna tracker with two servos. For mechanical reasons, I'm first mounting servo 1 on a base so that it tilts forward/backwards, then mount servo 2 on it twisted 90˚so it can tilt left/right.
I can basically use the first servo to select a from the great circles that go through $(az, el) = (0,0)$ and $(\pi, 0)$, and use the 2nd servo to move on the chosen great circle. This way, I should be able to point at the entire upper hemisphere even though I might need to reposition the antenna when crossing the midline (the servos only have 180 degrees of movement.)
I'm trying to find the function that maps az/alt to the tilt/tilt servo angles, which I think would be $(\epsilon, \lambda)$ in ecliptic coords (with $\beta =0$)
Looking at https://aas.aanda.org/articles/aas/full/1998/01/ds1449/node3.html i do find transforms for az/el to equatorial, and transforms from equatorial to ecliptic.
In my case, I think I want to use an ecliptic transform where I force the angle $\beta$ to be $0 $ and treat the ecliptic angle $\epsilon$ as a variable instead of constant.
Does that sound reasonable ? If so, is there any self-contained code around to implement these transforms, or even better is there a single transform that I can use ?
Thanks in advance!
EDIT:
I think I am a dolt. The equatorial - az/el transformation is a red herring. It's used in astronomy to map between the conceptual coordinate system centered in the middle of the earth to the one supporting a human sitting someone on top the planet with a telescope. With the antenna tracker, we don't have this problem, we just map between equatorial and ecliptic coordinates.
So the equations should look like
$\sin \beta = \sin \delta \cdot \cos \epsilon - \cos \delta \cdot \sin \epsilon \cdot \sin \alpha$
$\cos \lambda = (\cos \alpha \cdot \cos \delta) / \cos\beta$
$\sin \lambda = [\sin\delta \cdot \sin\epsilon + \cos\delta\cdot \cos\epsilon\cdot\sin\alpha] / \cos\beta$
Then I want $\beta=0$, so that the $\cos\beta$ terms become 1 and the $\sin\beta$ terms become 0, this simplifies to
$\epsilon=\arctan(\tan\delta\cdot\sin\alpha)$
$\lambda = \arccos(\cos\alpha\cdot\cos\delta)$
It would be great to get a confirmation that this is indeed correct (for some part of the sphere at least)