Globe

The Geometry of the Sphere




 John C.
Polking


Rice University 









The material on these pages was the text for part of the Advanced
Mathematics course in the High School Teachers Program at the IAS/Park City
Mathematics Institute at the Institute for Advanced Study during
July of 1996.  


Teachers are requested to make their own contributions to this page.
These can be in the form of comments or lesson plans that they have
used based on this material.  Please send email to the author at polking@rice.edu to inquire.  Pages can be kept
at Rice or on your own server, with a link to this page.


Putting mathematics onto a web page still presents a significant
challenge.  Much of the effort in making the following pages as nice
as they are is due to Dennis
Donovan.  Boyd
Hemphill added two nice appendices.  Susan Boone
helped construct the Table of Contents.  All of them are
teachers and members of the Rice
University Site of the IAS/Park City
Mathematics Institute.






Table of Contents






Introduction


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 that says that in any convex, three dimensional
polyhedron we have V - E + F =  2, where
V  is the
number of vertices, E  is the number of edges, and F
 is the number of faces.  









The next section
contains a 
discussion of the basic properties of the sphere.









The orthographic projection of the earth at the beginning of this
page, and the earth icon used throughout these pages were produced
with the program xearth,
written by  Kirk
Johnson of the University of Colorado.







John C. Polking <polking@rice.edu>



Last modified: Fri Jan 28 15:24:14 CST 2000