Hyperbolic Geometry and the Euclidean Parallel Postulate

Date: 01/20/99 at 11:15:45
From: Nelson
Subject: College geometry

In what type of geometry would this be true?

Given a line L and a point P, there is an infinite number of lines
passing through P parallel to L.

In this geometry, what is the sum of the angle measures of triangle?
Will it add up to 180 degrees?

Date: 01/20/99 at 12:55:01
From: Doctor Ken
Subject: Re: College geometry

Hi Nelson,

The type of geometry you're describing is called "hyperbolic
geometry." The definition of hyperbolic geometry is exactly the
property you state -- that given a line L and a point P not on L,
there are an infinite number of lines passing through P parallel to L.

Actually, most people use a different definition, which is the logical 
negation of the Euclidean parallel postulate. The Euclidean parallel 
postulate says that, for every line L and every point P not on L, 
there is exactly one line through P parallel to L.

The negation of that statement is: For some line L and some 
point P not on L, the number of lines through P parallel to L is 
either none, or more than one.

That statement is called the hyperbolic parallel postulate. One can 
show that given the other postulates of geometry, there must be at 
least one such parallel line. Then it's not too hard to see that it's 
equivalent to the property you gave. An excellent reference for this 
is Greenberg's book "Euclidean and Non-Euclidean Geometry."

For more information on the other questions you ask about hyperbolic 
geometry, search our archives for the terms "hyperbolic geometry."  
There's plenty there.

Good luck!

- Doctor Ken, The Math Forum
  http://mathforum.org/dr.math/