Figure fragment: The celestial sphere and its circumpolar sky regions

Declination and Altitude:

Declination (Dec,δ) is the angle measured from celestial equator toward the celestial axis. It is positive for the north direction and negative for the south direction. The declination range is δ∈[-90°,90°].
Declination and right ascension (RA) constitute equatorial coordinate system which is analogous to geographical coordinate system latitude and longitude.
Historically, high accuracy/precision declination measurements were done when astronomical objects transited (i.e., crossed) the celestial meridian. Measurements at other times, generally had more error and/or required using spherical trigonometry which is a nuisance.
It is of some interest to relate declination to the altitudes from due north and due south at the time of transiting the celestial meridian.
To do this easily, it is we must define declinations for due north and due south outside of the standard declination range δ∈[-90°,90°]: respectively, these are
```
  δ_dN = L + 90°  and  δ_dS = δ_dN - 180° = L - 90°  
```
where the first formula is obvious from Image 1 and the second formula from the first one.
The altitudes for general declination δ from due north and due south, respectively, are clearly
```
  A_N = -(δ - δ_dN) = -δ + L + 90°  and  A_S = (δ - δ_dS) = δ - L + 90°.  
```
The inverses of these formulae are
```
  δ = -A_N + L + 90°  and  δ = A_S + L - 90°  .
```
For latitude L, we have
```
  L = δ + A_N  - 90°  and  L = δ - A_S + 90°  .
```
Recall, latitude L is given as a negative value for southern latitudes.
The above formulae for the altitudes from due north and due south, their inverses and latitude L look a bit mysterious.
A concrete example can help to believe them. What is the altitude from due south for the Sun in Las Vegas, Nevada on the solstices and equinoxes? Behold:
```
 
   A_S = δ - L + 90°                  In general.
       = δ - 36.2° + 90° = δ + 53.8°  For L_LV ≅ 36.2°.
       = 23.4° + 53.8° = 77.2°        For the summer solstice (∼ Jun21).
       = -23.4° + 53.8° = 30.4°       For the winter solstice (∼ Dec21). 
       = 53.8°                        For the equinoxes (vernal ∼ Mar21, fall ∼ Sep21). 
```
Note, 23.4° is the famous tilt angle of the ecliptic (i.e., the great circle path of the Sun on the celestial sphere in a solar year = 365.2421897 days (J2000)) from the celestial equator. Note also, 23.4° is the Earth's axial tilt (currently 23.4°) (see also Wikipedia: Ecliptic: Obliquity of the ecliptic). On the summer solstice (∼ Jun21) (winter solstice (∼ Dec21)), the Sun is at its highest (lowest) point on the ecliptic and on the equinoxes (vernal ∼ Mar21, fall ∼ Sep21), the Sun is on the celestial equator.
An interesting special case is for the altitudes of the north celestial pole (NCP, δ=90°) and south celestial pole (SCP, δ=-90;). From the general altitude formulae above, we find:
```
  A_N_NCP = L  and  A_S_SCP = - L 
```
Recall, south latitudes are measured as negative. Note, negative altitude means the astro-body is below the horizon.
Another interesting special case is for the declination of the circumpolar circle at the zenith location. In this case, both A_N and A_S are 90° and we find δ = L for both.