Notes from Math 445 for February 9, 2004

Polygons on the Sphere

In Plane Geometry a triangle is the polygon with the least number of sides – 3.

In contrast spherical geometry allows for a 2-sided polygon.  This polygon is formed by 2 intersecting great circles and may be called a Bigon, Digon, or most often a Lune.

Is there a relationship between the area of a Lune and the angle between the great circles?

Let the area if the sphere = S

Q:  If one vertex angle of a line equals a, what is the other angle?

A: a

Q: What is the area of the lune if the vertex angle = a

A: aS/360

Why? It is equivilant to dividing the surface area into 360 congruent pieces each with a 1 degree angle and then taking however many are needed to create the lune based on the vertex angle.

            This relationship between area and angle exists in the sphere but there is no relationship possible in the plane.

	Consider:  You can get a series of isosceles triangles on the sphere by dividing a lune at different points. What do the resulting angles do? If you draw one lune using great circles you actually get 4 lunes.
	Question: If you create a triangle using another great circle and you know the angles of the triangle (a, b and c), is that enough to predict the area of the triangle? Hint: consider what you know about the lunes.