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Another way to slice the sphere...

Welcome to the first celestial navigation appendix! In this document, we will take an idea we saw in the Main Pages and apply it to a new way. Truth is, this is really an old way--it was used to find Latitude at sea long before the invention of the chronometer paved the way for the development of the altitude-intercept method. In addition, learning this technique is a superb way to understand the relationships between coordinate systems, so it will be well worth your time!

First, remember how we constructed our old friend, the Diagram on the Plane of the Equator. We imagined that we were looking at the Earth from high above the South Pole, and simply sketched what we would see from there; namely, our meridian, the Greenwich meridian, and hour circles of any celestial bodies circling the Earth. To make a Diagram on the Plane of the Meridian, we will do something similar.

Imagine that we are still a very long distance from the Earth, looking down upon it. This time, imagine that we are out somewhere above the Equator. Now how would we sketch what we see? As a circle again, of course! Only this time, the Earth appears to be sliced in half along an entire meridian, through both poles. When we draw the Diagram then, the North Pole is at the top of the circle, the South Pole at the bottom. The diameter of the circle between the 2 poles is therefore the Earth's axis. Then drawing-in the diameter perpendicular to the Earth's axis is clearly the Equator. The sketch looks like this:

  • Pn and Ps are the Poles

  • Q' and Q mark the intersections of the Equator with the meridian.

Although this diagram holds true for any meridian, in practical work we are concerned solely with what happens on our own meridian. So imagine that we are standing somewhere along the circle. For present purposes, we will presume ourselves to be standing in the Northern Hemisphere, on the right-hand branch of the meridian.

Now that we have that picture in mind, let us think of another very similar picture. Only in this case, instead of showing the Latitude-Longitude coordinate system of the Earth, let us instead depict the altitude-azimuth coordinate system of the Navigator. We will be slicing the entire celestial sphere in half along our principal vertical circle, more commonly called the celestial meridian by Navigators. Remember from the Page on coordinate systems that we are allowed to think of the principal vertical circle as a "meridian in the sky", or our meridian as a "vertical circle on the Earth" because they are both running from exactly North to exactly South and hence one lies exactly over the other. If we take our "god's-eye" view as being far to the West of the Earth, we get a picture like this:

  • Z is the zenith
  • N and S mark the intersections of the North and South points of the horizon with the celestial meridian.
  • The dashed line points down directly from the Navigator to the nadir.

Now a most remarkable thing happens if we consider that we can project our diagram of the Earth outward onto the sky. Specifically, we see that our meridian and our principal vertical circle exactly superimpose; or said another way, both the diagrams are really showing the exact same meridian!

This means we get to see relationships between the poles of the Earth, the Equator of the Earth, our Zenith, and our Horizon.

To illustrate, consider a familiar case--our original lighthouse example. This time, rather than standing at the bottom of a lighthouse, we will be standing exactly on the North Pole. So of course, our lighthouse beacon is then the North Star, Polaris.

We will sketch-in everything: our horizon, our zenith, the North Pole, the South Pole and the Equator. When we have finished, the picture, formally known as a Diagram on the Plane of the Meridian, looks like this:


  • Z is the zenith, which happens to be the North Pole.
  • N and S mark the intersections of the North and South points of the horizon, which just happens to be the Equator, with the celestial meridian.
  • The dashed line points down directly from the Navigator to the nadir, which happens to be the South Pole.

It is natural to ask at this point: What happens if we are not standing on the North Pole?

To answer this question, let us consider another extreme case. This time, we will go from the North Pole 90 degrees due South and stand on the Equator. Recall that important result from the Introductory lighthouse problem: the spot directly over your head is always directly over your head. With this in mind, we realize that now the Celestial Equator would be in the zenith, since we are standing on the Earth's Equator and we know that the Celestial Equator "matches" it exactly. So the Diagram on the Plane of the Meridian is easy to draw:

  • Z is the zenith, which happens to be the Celestial Equator.
  • N and S mark the intersections of the North and South points of the horizon, which just happen to be the Poles, with the celestial meridian.
  • The dashed line points down directly from the Navigator to the nadir, which happens to be the Equator on the "opposite side" of the Earth.

After these two examples, we can see a crucial point:

This leads to a crucial general rule:


The elevated Celestial Pole always has an Altitude equal to your Latitude.


(You may even notice that the word "Latitude" is an anagram on the word "Altitude"! Wow!!!)

This is a most important theoretical point we learn from the Diagram on the Plane of the Meridian. It is the basis of finding Latitude by observation of Polaris (for which tables may be found in the Nautical Almanac, but we will not consider it in these Web Pages.) But for practical work, we find a much more common application on the next Page...

Go to the Celestial Navigation Appendix on the "Noon Sight".

Go to the Celestial Navigation Appendices.

Go to the Entry Point to this tutorial