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### Spherical vs. Plane Geometry

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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.
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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/
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