The basic geometrical relationship between the Earth's surface and the sky

At any instant of time, each point in the sky has one point on the Earth directly below it. "Directly below" is precisely defined in terms of a straight line from the point in the sky to the center of the Earth; such a line passes through exactly one point on the Earth's surface (the point "directly below" the point in the sky). This point on the Earth's surface is called the "Geographical Position", or GP for short, of the point in the sky. As the Earth rotates, this relationship is constantly changing; the Nautical Almanac tabulates the GPs for a select number of celestial bodies at any given time, giving the GP coordinates in a form closely related to latitude and longitude.

 The diagram HP.JPG to the right shows an observer at o looking at a celestial body B, which is located directly above the point GP on the Earth's surface, and at a distance d from the Earth's center C. The observer, being on the Earth's surface, is a distance R (Earth's radius) from C. A line from C through o continues on to Z, the observer's Zenith. The line through o perpendicular to the line CZ represents the observer's horizon. The observer can use a sextant to measure the angle h. We desire to find the relation between the angle h and the angle H. If the distance d is much much larger than R (as is the case for the Sun, the stars, and all planets but Venus and Mars at their closest), the line of sight oB is practically parallel to the line CB, and simple geometry has h+H = 90 degrees. When d is not so large, which is especially true of the Moon, a more complicated relationship is needed, to account for so-called Horizontal Parallax.
 The diagram FIG4A.JPG to the right... 