The General Formulae for Declination-Altitude Conversions on the Meridian:
1. The general formulae for declination-altitude conversions on the meridian are relatively simple and have relatively simply derivations.
  The formulae and derivations are given Image 2.
2. Image 2 Caption: To recapitulate the general formulae from Image 2:
  1. A_N/S = (±)_N/S(L - δ) + 90°
  2. δ = L +(±)_N/S(90° - A_N/S)
  3. L = δ +(±)_N/S( A_N/S - 90°)
  where:
  1. δ is declination.
  2. A_N/S is altitude (upper case N or + is measured from due north and lower case S or - measured from due south.
  3. L is latitude.
  4. Nota bene: we count south latitude as negative.
3. What if you want to determine declinations from positions NOT on the meridian. The you need to use spherical trigonometry and the necessary formulae and derivations are quite complex. However, nowadays the computer will do that for you.
4. What are the special case formulae for when the altitude is for the north celestial pole (NCP) (δ_NCP = 90°) and the south celestial pole (SCP) (δ_SCP = -90°)?
  From the general formulae:
  1. A_{N_NCP} = L.
  2. A_{S_NCP} = -L + 180°. For example, say you are at L = 45°, then the altitude of the NCP is 135° from due south which is obviously correct.
  3. A_{N_SCP} = L + 180°. For example, say you are at L = -45°, then the altitude of the SCP is 135° from due north which is obviously correct.
  4. A_{S_SCP} = - L. This is the mirror image of the special case A_{N_NCP} = L. Note if you are in the Southern Hemisphere and count ount south latitude as positive (which is the normal convention), then the mirror image case is exactly as expected A_{S_SCP} = L(south counted positive).
5. In days of yore, measurements of declinations from transits of the meridian were very important in astrometry (i.e., positional astronomy) and celestial navigation since they could be done easily without spherical trigonometry.
  To determine declinations from positions NOT on the meridian. The you need to use spherical trigonometry and the necessary formulae and derivations are quite complex. However, nowadays the computer using spherical trigonometry easily converts horizontal coordinates to equatorial coordinates (and vice versa) and measurements at transits of the meridian are much less important than they were.

The Circumpolar Sky as A Function of Latitude:

In the Northern Hemisphere, the circumpolar sky above/below the horizon must be closer to the NCP/SCP than an angle equal to A_N_NCP = L. From viewing Image 1, clearly the limiting declinations for the circumpolar sky obey
δ_(±)= ±(90° - L) ,
where the +/- case is lower/upper limit for declinations above/below the horizon.
What of the Southern Hemisphere? Hm, tricky. But we can make use of the mirror reflection symmetry of the celestial equatorial plane to write
```
     -δ_(above/below)= ±[(90° - (-L)]
     -δ_(below/above)= ∓[(90° - (-L)]
     δ_(below/above)= ±[(90° - (-L)]
     δ_(±)= ±[(90° - (-L)]  .  
```
where again southern latitudes are counted as negative and the +/- case is lower/upper limit for declinations below/above the horizon.
The general formula is clearly
δ_(±)= ±(90° - |L|) ,
where the +/- case is lower/upper limit for circumpolar sky declinations, and +/- case is above/below the horizon for L > 0 and +/- case is below/above the horizon for L < 0.
At the equator (L=0), there is NO circumpolar sky. At the poles (L=±90°) all the sky is circumpolar sky.

One can write down some useful formulae for the circumpolar sky:

  f_c = 1 - cos(L)          :  fraction of the sky that is circumpolar
  f_s = (1/2)*(1 - cos(L))  :  fraction of the sky that is circumpolar
                                 and above/below the horizon
  f_n = cos(L)              :  fraction of the sky that is non-circumpolar

which follow from the integral

  [1/(4π)]*2*2*π* ∫_μ¹dμ = 1 - μ = 1 - cos(90-(90-|L|)) = 1 - cos(L)  .

The fraction of sky that is non-circumpolar can all be seen from any latitude, but only half at any time in the day. The Earth occults the other half. The rotation of Earth allows you to see the whole non-circumpolar sky over the course of a day.

The above formulae for the fractions of sky have been used to construct Table: The Circumpolar Sky given below.

  _____________________________________________________________________

  Table:  The Circumpolar Sky 
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  Latitude N/S      f_c      f_s      f_n   f_n exact   comment
    (degrees)
  _____________________________________________________________________

         0       0.0000   0.0000   1.0000           1   on equator
        30       0.1340   0.0670   0.8660   sqrt(3)/2
        36       0.1910   0.0955   0.8090         ...   in Las Vegas
        60       0.5000   0.2500   0.5000         1/2
        90       1.0000   0.5000   0.0000           0   at North Pole
                                                          or South Pole
  _____________________________________________________________________
  Reference:  CAC-8--9
  _____________________________________________________________________

AN/S = (±)N/S(L - δ) + 90°

δ = L +(±)N/S(90° - AN/S)

L = δ +(±)N/S( AN/S - 90°)

δ_(±)= ±(90° - L) ,

δ_(±)= ±(90° - |L|) ,

A_N/S = (±)_N/S(L - δ) + 90°

δ = L +(±)_N/S(90° - A_N/S)

L = δ +(±)_N/S( A_N/S - 90°)