Spherical vs. Plane Geometry

Date: 05/30/97 at 22:08:52
From: Derrick Tsang
Subject: Spherical geometry

Hello,

I am a high school student and I need to compare and contrast the 
characteristics of plane and spherical geometry.  If someone could 
help me, it would be much appreciated.

Thank you.

Date: 05/31/97 at 03:04:34
From: Doctor Mitteldorf
Subject: Re: Spherical geometry

Dear Derrick,
   
In plane geometry, for example, the straight line is the shortest 
distance between two points. In contrast, on a sphere (think of the 
earth's surface) the shortest distance between two points is a "great 
circle."
   
In a plane, the circumference of any circle is 2*pi*r. But suppose we 
imagine traveling r miles along the earth's surface from the north 
pole, then drawing a circle (a parallel of latitude, really) around 
the pole from that point. The circumference of that circle would be 
less than 2*pi times the distance we traveled.
   
Draw any triangle in the plane, and the sum of its angles is 180 
degrees. But suppose we draw a "triangle" on the earth's surface, 
where each line in the triangle is not really a line but a great 
circle, since that's the closest thing we have to straight lines.  
If the triangle is small, the sum of its angles is about 180 degrees.  
But if the triangle is large, the sum of the angles can be much 
larger. In fact, you can draw a "triangle" with three right angles in 
it on the surface of the earth!
   
This should get you started. Think of some more things that are true 
in a plane but are true on a sphere only for very small figures.

-Doctor Mitteldorf,  The Math Forum
 Check out our web site!  http://mathforum.org/dr.math/