Euclidean geometry has 5 postulates

1. Any two points can be joined by a straight line.

2. Any straight line segment can be extended indefinitely in a straight line.

3. Given any straight line segment, a circle can be drawn having the segment as radius and one endpoint as center.

4. All right angles are congruent.

5. Parallel postulate. If two lines intersect a third in such a way that the sum of the inner angles on one side is less than two right angles, then the two lines inevitably must intersect each other on that side if extended far enough.

It was by trying to prove the 5th postulate false that lead mathematician's to discover hyperbolic geometry. In fact Jesuit Girolamo Saccheri (1733) I guess is considered the father of hyperbolic geometry, he was trying to do just that.

All through High School and College I never considered that trig was really about objects on a 2-D plane. You plot points on a cartisian x-y plane, and draw your f(x) or object, but that really does not represent real world, because we live in a 3-D world, plus time, so you can consider that we are living in 3-D, space-time existance. From what little I understand of hyperbolic geometry, that is a more accurate representation of the universe that we live in.

a^2 + b^2 = c^2

Normally you have a problem in a trig math book were you figure out how high the plane is in the air. This formual does not take in to account the curvature of the earth, so what you are doing is finding a close approximation of how high the plane is above the earth. Also the plane is following a curved path, not straight. I need to learn more about hyperbolic geometry.