the geometry of the sphere
We all know that the shortest path between two points on a plane is a
straight line. On the surface of a sphere, however, there are no
straight lines. The shortest path between two points on the surface of
a sphere is given by the arc of the great circle passing through
the two points. A great circle is defined to be the intersection with a
sphere of a plane containing the centre of the sphere. If the plane does
not contain the centre of the sphere, its intersection with the sphere is
known as a small circle. In more everyday language, if we take
an apple, assume it is a sphere, and cut it in half, we slice
through a great circle. If we make a mistake, miss the centre
and hence cut the apple into two unequal parts, we will have sliced through
a small circle. Figure 1 illustrates the concepts of great and small
upper panel: examples of great circles. lower panel:
examples of small circles. The black dot
represents the centre of the sphere.
Another concept we should all be familiar with from planar geometry is the
triangle. If we wish to connect three points on a plane using the shortest
possible route, we
would draw straight lines and hence create a triangle. By analogy, if we
wish to connect three points on the surface of a sphere using the shortest
possible route, we would draw arcs of great circles and hence create a
spherical triangle. To avoid ambiguities, a triangle drawn on
the surface of a sphere is only a spherical triangle if it has all
of the following properties:
Hence, in Figure 2, triangle PAB is not a spherical triangle
(as the side AB is an arc of a small circle), but triangle
PCD is a spherical triangle (as the side CD is an arc of
a great circle). You can see that the above definition of a spherical
triangle also rules out the "triangle" PCED
as a spherical triangle, as the vertex angle P is greater than
180° and the sum of the sides PC and PD is less than
- The three sides are all arcs of great circles.
- Any two sides are together greater than the third side.
- The sum of the three angles is greater than 180°.
- Each spherical angle is less than 180°.
Triangle PCD is a spherical triangle. Triangles PAB
and PCED are not spherical triangles.
It is important to realise that the
lengths of the sides of a spherical triangle
are expressed in angular measure.
This follows from the fact that the length of an arc s which subtends
an angle at the centre of a circle of radius r
is given by
s = ( / 360) . 2 r
where is measured in degrees and s is given in
whatever length units are used for r.
Recalling that 2 radians = 360°, this gives
s = r
where is now measured in radians
and s is still given in whatever length units are used for r.
Given that spherical triangles are all made up of great circle arcs, and all
great circle arcs on a sphere are of the same radius, it is convenient to
take the radius of the sphere as unity and then write
which shows that the length of a side of a spherical triangle
is equal to the angle (in radians) it subtends at the centre of the
sphere. For example, if s is one-quarter of the circumference
of a great circle, s is then / 2 radians and there
is no ambiguity if it is then expressed as 90°.
©Vik Dhillon, 30th September 2009