A_{N_NCP} = L .

Note that a right angle can be rotated into a right angle.

Also note that B = L by the converse of Euclid's parallel postulate.

1. A Bit More Detail on the Formula for A_{N_NCP}:
   1. Image 1 is a cross-sectional view of the Earth.

   2. The Earth's axis is extended to the north celestial pole (NCP) and the Earth's equator is projected outward to the celestial equator from the Earth's center.

   3. The NCP and celestial equator are on the celestial sphere which is infinitely remote from the Earth which is like a point relative to the celestial sphere.

   4. All lines starting from the Earth toward the NCP or celestial equator are effectively parallel---precisely because the Earth is like a point relative to the celestial sphere.

   5. L is value of the latitude of a general observer on the Earth's surface in the Northern Hemisphere. It's NOT 45° or any specific angle---it just looks that way.

   6. The observer is tiny, and so the Earth's surface to them is the infinite horizon plane which cuts the celestial sphere in half. The cut line is a great circle and is, in fact, the horizon itself.

   7. Image 1 clearly shows for a general point on the Earth's in the Northern Hemisphere that we obtain the above formula A_{N_NCP} = L.

   8. Actually, the same formula holds for the Southern Hemisphere if you count south latitudes as negative. We prove this statement in section The General Formulae for Declination-Altitude Conversions on the Meridian which appears below or in the extended version of this figure (i.e., Celestial sphere file: declination_altitude.html), where the general formulae for declination-altitude are given and derived and where the circumpolar sky as a function of latitude is explicated in section The Circumpolar Sky as A Function of Latitude.

   9. Note in the Southern Hemisphere the NCP is below the horizon, and so its altitude is negative.