Re: Spherical Geometry

From: Michael Orion (beeworks_at_hotmail.com)
Date: 10/08/04 

Date: Fri, 8 Oct 2004 14:25:16 +0000 (UTC)

On 01 Jun 1997, Derrick Tsang wrote:
><pre>
>Hello I have to create a report by comparing and contrasting plane and
>spherical geometry. COuld someone help me with some examples please?
>It would be much appreciated thanx!
>
>
>GSMILES@worldnet.att.net
>
>
></pre>

Derrick,

Here are a couple of nice things to compare:

1) Sum of Anlges: Let sigma = sum of the angles. Then in the plane sigma = pi, whereas on the sphere sigma > pi. Extra: what is the upper bound on sigma on the sphere?

2) Area of the triangle: In the plane it is A = (1/2)bh, where b is the base and h the height. On the sphere it is A = sigma - pi, where sigma is the sum of the angles.

3) Law of sines: On the plane it is a/sin(alpha) = b/sin(beta) = c/sin(gamma), where alpha (beta, gamma) is the angle opposite the side a (b, c). On the sphere it is sin(a)/sin(alpha) = sin(b)/sin(beta) = sin(c)/sin(gamma).

4) Pythagorean Theorem: On the plane it is a^2 + b^2 = c^2, on the sphere it is cos(a)cos(b) = cos(c).

5) Law of cosines (generalization of Pythagoream Theorem): On the plane it is c^2 = a^2 + b^2 - 2(ab)cos(gamma), where a, b, c are the sides of the triangle and gamma is the anlge opposite c. On the sphere it is cos(c) = cos(a)cos(b) + sin(a)sin(b)cos(gamma). Extra: the law of cosines for angles on a sphere is
cos(gamma) = -cos(alpha)cos(beta) + sin(alpha)sin(beta)cos(c)

Hope these little tid-bits help.

- MO