What is a specific topic that I can expand upon in non-euclidean geometry?

Question

I looking for some sort of question that I can do that has to do with its real world functions

Answer 1

hyperbolic plane, you can check what the different models for hyperbolic geometry are.

anotherone, would be Riemannian geometry, which is the basis for relativity theory.
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Answer 2

Parallel lines.....

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Visualizing hyperbolic geometry

M. C. Escher's famous prints Circle Limit III and Circle Limit IV illustrate the conformal disc model quite well. In both one can see the geodesics (in III the white lines are not geodesics, but hypercycles, which run alongside them). It is also possible to see quite plainly the negative curvature of the hyperbolic plane, through its effect on the sum of angles in triangles and squares.

For example, in Circle Limit III every vertex belongs to three triangles and three squares. In the Euclidean plane, their angles would sum to 450°, i.e. a circle and a quarter. From this we see that the sum of angles of a triangle in the hyperbolic plane must be smaller than 180°. Another visible property is exponential growth. In Circle Limit IV, for example, one can see that the number of angels within a distance of n from the center rises exponentially. The angels have equal hyperbolic area, so the area of a ball of radius n must rise exponentially in n.

There are several ways to physically realize a hyperbolic plane (or approximation thereof). A particularly well-known paper model based on the pseudosphere is due to William Thurston. In 1997, Daina Taimina crocheted a hyperbolic plane based on Thurston's models. In 2000, Keith Henderson demonstrated a quick-to-make paper model dubbed the "Hyperbolic soccerball."



Applications Of Spherical Geometry

Spherical Geometry is also known as hyperbolic geometry and has many real world applications. One of the most used geometry is Spherical Geometry which describes the surface of a sphere. Spherical Geometry is used by pilots and ship captains as they navigate around the world. However, working in Spherical Geometry has some nonintuitive results. For example, did you know that the shortest flying distance from Florida to the Philippine Islands is a path across Alaska? The Philippines are South of Florida - why is flying North to Alaska a short-cut? The answer is that Florida, Alaska, and the Philippines are collinear locations in Spherical Geometry (they lie on a "Great Circle"). Small triangles, like ones drawn on a football field have very, very close to 180 degrees. Big triangles, however, (like the triangle with veracities: New York, L.A. and Tampa) have much more then 180 degrees.


ALSO:
http://www.mccallie.org/myates/Taxicab%20Geometry/taxicab.htm

Answer 3

Probably the easiest application of non-Euclidean geometry that is practical in the real world is using spherical geometry to compute shortest-distance paths (great circles) between points on the globe. Airlines need to deal with this kind of geometry all the time.

Answer 4

Euclid's Parallel Postulate is certainly a possibility. In the Geometry of the sphere and pseudosphere, two lines may always intersect or may always be parallel. It is probably the most disputed of all Euclid's postulates.

Another possibility is the sum of the angles in a triangle. Euclid saya 180 degrees. Another Geometry (Gauss) says they are always less than 180 degrees and another says they are greater than 180 degrees.

Look up B. Riemann, N. Lobachevsky, J. Bolyai, and C.F. Gauss for pioneers in this subject.