Astronomy Principles and Practice Fourth Edition.pdf

46 The geometry of the sphere
Figure 7.1. The basis of spherical geometry.
By construction, F is any point on figure EFGHE so that all points on EFGHE must be
equidistant from K . Since they also lie on a plane, EFGHE must be a circle, centre K .
P and Q are also the poles of circle EFGHE.
Draw another great circle PGCQP through the poles P and Q to intersect the small circle
EFGHE and great circle ABCDA in G and C respectively.
Then the angle between the tangents at P to the great circles PFBQP and PGCQP is said to
be the spherical angle at P or angle GPF or angle CPB. A spherical angle is defined only with
reference to two intersecting great circles.
If three great circles intersect one another so that a closed figure is formed by three arcs of the
great circles, it is called a spherical triangle provided that it possesses the following properties:
1. Any two sides are together greater than the third side.
2. The sum of the three angles is greater than 180 ◦ .
3. Each spherical angle is less than 180 ◦ .
In figure 7.1, therefore, figure PBC is an example of a spherical triangle but figure PFG is not,
the latter being excluded because one of its sides (FG) is the arc of a small circle. It may be noted in
passing that △PBC is a special case where two of the angles, namely PBC and PCB, are right
angles.
The sides of a spherical triangle are expressed in angular measure.
The length s of the arc BC is given in terms of the angle θ it subtends at the centre of the sphere
and the sphere’s radius R by the relation
s = R × θ
where θ is expressed in radians.
It should be remembered that
Other useful relationships are:
2π radians = 360 degrees.
1radian≈ 57·3 degrees (57·◦3)
≈ 3438 arc minutes (3438 ′ )
≈ 206 265 arc seconds (206 265 ′′ ).