Navigational triangle

The navigational or PZX triangle is a term used in Astro-Navigation, or Celestial Navigation, and its solution can give you your position anywhere (more or less) on the globe. The sky is described as a Celestial Sphere with a North and South Pole corresponding to our own North and South Pole. Conveniently, the 'Pole Star' lies (more or less) at the North Pole of the Celestial Sphere.

The PZX triangle is a spherical triangle with the following points of reference:

P is the Celestial Pole (North or South). For Astro Navigational use, it remains a fixed point.

Z is the point on the observer's meridian, where a line from the centre of the earth (treated in this case as a 'sphere') projected outwards through the observer and onto the Celestial Sphere points to his Zenith. A simple way of thinking about it is to imagine that when you are standing on Z, your feet point directly towards the centre of the earth, and your head points to your Zenith. (A line in the opposite direction, through your feet and the centre of the earth, would point to your 'Nadir') Thus 'Z', the observer's position on the earth, can be described in terms of 'Latitude' (degrees north or south of the Equator) and Longitude (degrees east or west of the Greenwich Meridian).

X is the point on the Celestial Sphere of any heavenly body: The Sun, The Moon, The Planets, The Stars. Some of these points appear to move relative to the celestial sphere (The Sun, The Moon, The Planets), but the Stars (thankfully) remain relatively fixed (at least as far as Astro Navigation is concerned). X is described in terms of its angular height above or below the equator, known as its Declination (corresponding to its Latitude), and its 'Hour angle', the angle between its meridian and the Greenwich meridian (corresponding to its Longitude). This angle, known as the Greenwich Hour Angle', gives its Longitude. If you are standing on the Greenwich meridian, the angle between your meridian and that of X will be its Greenwich Hour Angle, measured in degrees. However, if you are somewhere else, then the angle between you and the object is known as its Local Hour Angle.

Any of these points on the Celestial Sphere can be joined by an imaginary line to the (theoretical) centre of the earth, and the point at which they meet the surface of the earth gives you a spherical triangle with the vertices P,Z and X.

If you know, or can discover, the Angles subtended by P,Z and X at any given point in time, then you can fix your position anywhere on the globe.

The angle subtended at P between Z and X is the Local Hour Angle (LHA)of X. If Z is on the Greenwich Meridian, then that Angle must be the Greenwich Hour Angle (GHA) of X. If Z is not on the Greenwich Meridian, then you have the Local Hour Angle (LHA) of X, and if you know the GHA of X, all you need to do is add subtract the LHA of X accordingly, to discover the Longitude of Z. As you are usually standing on the point Z, you have your Longitude. If the sum comes to more than 360 degrees, simply subtract 360, the answer is simple.

Finding your Latitude is slightly more complex. You need to be able to measure the vertical angle of X from your horizon, and for this you need a Sextant. If you know the Declination of X (you can find this in a reference book known as an Almanac), then with the use of a set of tables known as 'Sight Reduction Tables' and some simple mathematics, you can work out your Latitude as accurately as you can measure the vertical angle, or Altitude, of X.

References

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