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/
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