The Geometry of the Sphere
id = 926
, parent = 925
, thread = 923
, catid = 26
, locked = 0, moved = 0,
userid = kyanh
, ip = 220.127.116.11, time = 2006/01/12 (1137056104) ,
subject = Re:hyperbolic geometry
, hits = 0, karma = 0+0-,
We are interested here in the geometry of an ordinary sphere. In plane geometry we study points, lines, triangles, polygons, etc. On the sphere we have points, but there are no straight lines --- at least not in the usual sense. However, straight lines in the plane are characterized by the fact that they are the shortest paths between points. The curves on the sphere with the same property are the great circles. Therefore it is natural to use great circles as replacements for lines. Then we can talk about triangles and polygons and other geometrical objects. In these notes we will do this, and at the same time we will continuously look back to the plane to compare the spherical results with the planar results.
We will study the incidence relations between great circles, the notion of angle on the sphere, and the areas of certain fundamental regions on the sphere, culminating with the area of spherical triangles. Our ultimate goal is two very nice results. First we will prove Girard's Theorem, which gives a formula for the sum of the angles in a spherical triangle. Then we will use Girard's Theorem to prove Euler's Theorem...