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Summary: given a formula(s) that shows a relation between several variables, how to make Mathematica find all relations between these variables.

raDecLatLonGMST2azAlt[ra_, dec_, lat_, lon_, gmst_] = 
 {ArcTan[Cos[lat]*Sin[dec] - Cos[dec]*Cos[gmst + lon - ra]*Sin[lat], 
  -(Cos[dec]*Sin[gmst + lon - ra])], 
 ArcTan[Sqrt[(Cos[lat]*Sin[dec] - Cos[dec]*Cos[gmst + lon - ra]*Sin[lat])^2 + 
    Cos[dec]^2*Sin[gmst + lon - ra]^2], 
  Cos[dec]*Cos[lat]*Cos[gmst + lon - ra] + Sin[dec]*Sin[lat]]}

The function above takes 5 inputs (ra, dec, lat, lon, and gmst) and yields two outputs. The outputs aren't formally named, but I refer to them as az and alt, in that order.

Symbolically, this could be considered a formula taking:

{ra, dec, lat, lon, gmst} -> {az,alt}

or, since az and alt are independent, as two separate formulas (which might be easier to deal with):

{ra, dec, lat, lon, gmst} -> az
{ra, dec, lat, lon, gmst} -> alt

Of course, there are several other relationships between the 7 variables (ra, dec, lat, lon, gmst, az, and alt). For example:

  • given {ra, dec, lat, lon, alt}, it's possible to find gmst

  • given {dec, lat, alt}, it's possible to find az (there are actually two values, more if you allow for periodicity)

My question: how can I have Mathematica give me all possible formulas that take a subset of these values as input and yield a subset of these values as output?

I can do this manually to some extent using Solve, but the combinations are tedious, and I was hoping there was an automatic method.

Note the example formulas given is from astronomy, but my question is generic to any multi-variable formula.

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    $\begingroup$ Perhaps you could automate the search by running Solve with appropriately chosen variables to solve for / to eliminate. Maybe start from the Subsets of your set of variables? $\endgroup$ – MarcoB May 3 '18 at 21:00
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Your equation is very complicated... here's a simple example of what you are asking for -- a function of 3 variables is to be solved in turn for each of the variables:

f[x_, y_, z_] := x - y^2 + Sqrt[z];
Solve[f[x, y, z] == 0, #] & /@ {x, y, z}

{{{x -> y^2 - Sqrt[z]}}, 
{{y -> -Sqrt[x + Sqrt[z]]}, {y -> Sqrt[x + Sqrt[z]]}}, 
{{z -> x^2 - 2 x y^2 + y^4}}}
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