Class notes February 9, 2004

Polygons on a sphere –

2-sided polygons exist on a sphere, they are called lunes. They are also referred to as bigons or digons.

2 intersecting great circles define 4 lunes.

If we know the angle between the intersecting great circles, can we find the area of the lune? (Given the surface area of the sphere = s)

Q1: If 1 vertex angle of a line = a, what is the other angle? = a.

Q2: What is the area of the lune if vertex angle = a? s x (a/360)

This can be thought of as dividing the sphere into 360 1degree slivers then adding “a” of these slivers together.

So there is a relationship between area and angles.

Drawing a third great circle, excluding the “equator”, and any that go through the existing vertex, we get a spherical triangle. The sphere is divided into eight pieces.

2 spherical triangles make a lune.

Can we determine what is the area

Of triangle abc?