Spherical trigonometry

A spherical triangle is a triangle drawn on the surface of a sphere. Spherical trigonometry is the application of a set of techniques which find the remaining angles and sides of a spherical triangle when only some of these are known. (It is possible to consider spherical rectangles too, but that topic is beyond the scope of this page.)

We will not be concerned here with the derivations of the various formulae. You will find these in books devoted to the topic. Instead, we will see how a home made universal astrolabe. can be used to solve simple problems of spherical trigonometry. The program which accompanies this page can be used to construct such a device. You will need to print out a plate on paper (Menu: Univ.Astr. / Plate general) and a rete on transparent film (Menu: Univ.Astr. /Rete general). I recommend that you make the brachiolus (a pointer) by printing the appropriate components on transparent film. Some parts must be pivotted. After making suitably sized holes, I fasten together any such parts using a press-stud (the things which are used to fasten together parts of ladies clothing, and which make a click as they are fastened).

If you haven't made a universal astrolabe yet, start now!

However, if you still have to buy the transparent material, you can probably manage temporarily with an interactive display of the universal astrolabe on the screen (Menu: Univ.Astr. /Summer solstice). Ignore the stars and the ecliptic circle. You can use the latitude buttons to set the rotation of the rete above the plate, but remember that the latitude setting rotates the rete by an angle equal to 90 degrees minus the latitude setting.

Introduction to spherical trigonometry

1. A great circle is a circle on the surface of a sphere whose centre is at the centre of the sphere. The equator and the lines of longitude around the earth are examples of great circles. (On this page, the earth is considered to be a sphere.) The circles which indicate latitude are small circles, as are the circles showing the Tropics of Capricorn and Cancer. Small circles do not have the centre of the earth at their centre. Do not confuse small circles with great circles.

2. A spherical triangle is a 'triangle' on the surface of a sphere whose three sides are arcs of great circles.

3. The length of each side is the length of the arc, and is measured in degrees, this being the angle which the points at the ends of the arc make at the centre of the sphere.

4. There are three angles between these three sides. The sum of these three angles does not make 180 degrees. Instead, the total is always greater than 180 degrees.

5. When a spherical triangle is illustrated, each of the three sides is drawn as an arc.

As an example, a spherical triangle drawn between
  the North Pole,
  the equator at 0 degrees longitude and
  the equator at 90 degrees longitude
will contain three angles in the corners each of 90 degrees, whose total is 270 degrees. As well as the three corner angles being 90 degrees, the lengths of the three sides are also 90 degrees in this example.

Similarly, a triangle drawn between
  latitude 50 degrees at 0 degrees longitude,
  the equator at 0 degrees longitude and
  the equator at 40 degrees longitude
will have sides of
  50 degrees,
  40 degrees and
  60 degrees (approx).
In the corners, the triangle will have angles of
  90 degrees (at the equator, 0 degrees longitude),
  about 48 degrees and
  about 62 degrees.

Solving a spherical triangle problem

First you must know how a spherical triangle appears when you identify it on a universal astrolabe. We will start with a triangle having sides of 40, 50 and 60 degrees. We will then see that the three angles in the corners have values of approximately 48, 62 and 90 degrees.

After starting with the pole of the rete superimposed over the pole of the plate, rotate the rete clockwise through the angle of the longest side, which in our example is 60 degrees. The pole on the rete will now be separated from the pole of the plate by an angle of 60 degrees. The arc around the outside circle between the two poles shows us one side of our triangle: the side which has a length of 60 degrees.

Counting from the pole of the plate, count parallel arcs on the plate to find the 50 degree parallel arc. Measured from the pole of the rete, count parallel arcs on the rete to find the 40 degree parallel arc. Arrange a pointer to point to the crossing point of these two arcs. On the plate, find the polar arc which crosses through this point. The section of this arc from the pole on the plate to the crossing point shows you the second side of the triangle. On the rete, find the polar arc which crosses through this point. The section of this arc from the pole on the rete to the crossing point shows you the third side of the triangle. Thus, you can now 'see' your triangle on the astrolabe, shown by these three arcs.

On the plate, if you count the polar arcs from the outside edge to the crossing point, you will have found the value of the angle which is at the corner of the triangle next to the pole of the plate. Similarly, on the rete, if you count the polar arcs from the outside edge to the crossing point, you will have found the value of the angle which is at the corner of the triangle next to the pole of the rete.

This technique doesn't show you the angle which is opposite the side represented by the arc between the two poles. To find the value of the angle in the third corner, you must repeat the exercise, using a different side for the rotation of the rete.

If you look at the triangle you have revealed once more, you will see that the lengths of the sides can be determined by counting the polar arcs which cross them, and the angles in the two corners adjacent to the poles can be determined by counting the arcs which radiate within them.

Unfortunately, there may be a complication. If you know the lengths of two sides and an angle which is not the angle between them, there may be two triangles which have these requirements. Similarly, if you know two angles and the length of a side which is not between the two angles, there may again be two triangles which have these requirements. This difficulty is described in books as 'ambiguity'. This means that when you are looking for the crossing point of two arcs, you will find that one arc crosses the other in two positions. Thus, you have a choice between two triangles. In practice, you can usually examine both triangles on your astrolabe and can then select the one which is applicable.

To summarise, you can use variations of this technique to identify any spherical triangle with three knowns, and can then determine two of the three unknowns.

Only one small point remains. If you are identifying a right-angled triangle, it makes sense for the hypotenuse to be the angle between the pole of the plate and the pole of the rete. The right-angle is then at the point you locate, and is the angle which doesn't have radiating arcs within it.

Spherical astronomy

Spherical astronomy is spherical trigonometry with the sides and angles of the most commonly used triangle in astronomy renamed to have appropriate astronomy terms. These astronomy terms are related to the celestial sphere and the coordinates of the observer's view.

The three points at the corners of the spherical triangle are commonly marked
  P, Z and O, standing for
  the celestial Pole,
  the observer's Zenith, and
  the Object of interest (a star or the Sun).
  The angle at P is the hour angle,
  the angle at Z is the azimuth, and
  the angle at O is usually of no interest.
  The side opposite P is named the co-altitude,
  the side opposite Z is named the co-declination, and
  the side opposite O is named the co-latitude.

(The co-altitude is 90 degrees minus the altitude,
the co-declination is 90 degrees minus the declination,
and the co-latitude is 90 degrees minus the latitude.)

On a universal astrolabe, the spherical triangle has at its corners
  the pole on the plate,
  the pole on the rete and
  the position of the star (or the Sun).
The triangle is formed by
  the arc between the two poles,
  the polar arc on the plate from its pole to the star and
  the polar arc on the rete from its pole to the star.

Let us consider a star, Aldebaran, whose RA is 4 hours 36 minutes and whose declination is 16 degrees 31 minutes (year 2000). A quick calculation converts this to
  an RA of 66 degrees and
  a declination of 16.5 degrees, approximately.
Suppose you want to determine the azimuth and elevation of Aldebaran, at midnight during the night of the Summer Solstice.

You will call the pole of the plate the celestial pole, the pole of the rete the zenith, and rotate the rete by an angle equal to your co-latitude. As you do that, you will probably be thinking, "This is all easy to remember because those are the definitions usually given to those points on the plate and rete of a universal astrolabe."

Conveniently, the position of Aldebaran is already marked on the plate. If you check this position, you will find that its co-declination (90 - 16.5 is 73.5) is already set for our triangle. Thus, counting parallel arcs to it from the pole on the plate, you will find that its co-declination is set at about 73.5 degrees. Similarly, its hour angle at the due time and date is already set at 1 hour 24 minutes (19 degrees) before midnight, which you can determine from the polar arcs of the plate, counting from the right-hand edge of the astrolabe to the star.

So what is the azimuth and declination of Aldebaran? Counting polar arcs around the zenith point on the rete, from the pole of the plate to the star, you will find the azimuth. Counting parallel arcs from the zenith point on the rete, from the pole of the rete to the star, you will find the co-elevation. Yes, that is almost precisely what you do when you use a universal astrolabe normally. The almost insignificant difference in procedure is that you count up to the elevation of the star from the horizon, rather than count the co-elevation from the pole.

So at midnight during the night of the Summer Solstice, after setting your latitude you already know how to use a universal astrolabe to determine the position in the sky of a star of known ra and declination. Without realising it, when you have previously been using the universal astrolabe to find its azimuth and elevation, you have used it to solve a spherical triangle problem of spherical astronomy.