Who is asking: Student
Level: Secondary
Question:
How can you use non-euclidean geometry to navigate on a sphere? What
geometers did work in this area?
Hi Geoffrey,
There are two closely related but quite different branches of geometry that are based on the sphere. One is called SPHERICAL GEOMETRY. It is a part of 3-dimensional Euclidean geometry that was developed by the ancient Greeks more than 2000 years ago. The theory was applied, particularly by Ptolemy (around 100 AD), to study the stars and planets. Spherical geometry is what one uses to navigate on a sphere. It is our familiar Euclidean geometry, with lots of trigonometry involved.
The other relevant geometry is called ELLIPTIC GEOMETRY. It is one type of classical non-Euclidean geometry, developed mostly by German mathematicians in the last half of the 19th century based on ideas of Bernhard Riemann (1826-1866). It differs from spherical geometry by considering points on the opposite sides of a sphere to be the same point. One can use the formulas of this non-Euclidean geometry only to navigate over portions of the earth, but it would be very inconvenient to travel halfway around the world to find yourself back where you started. In short, it is more convenient to use Euclidean geometry to answer questions about a Euclidean universe.
You can learn more about geometry by consulting
http://mathworld.wolfram.com/
Chris