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.

**A Bit More Detail on the Formula for A**_{N_NCP}:- Image 1 is a cross-sectional
view of the Earth.
- 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.
- 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.
- 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.
- 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. - 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.
- 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 .By mirror reflection, we obtain the analogous formula for the altitude of the SCP from due south: are, respectively,

A

_{S_SCP}= L .Note we count south latitudes as positive as we usually do. However, counting them as negative allows for more general formulae to be obained relating declination (dec or δ), altitude, and latitude.

Note also in the Northern Hemisphere/Southern Hemisphere SCP/ the NCP is below the horizon, and so its altitude is negative.

- We prove more general formulae relating
declination (dec or δ),
altitude,
and latitude.
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_4.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.

- Image 1 is a cross-sectional
view of the Earth.