what do terms like lines and angles mean after Euclid's parallel postulate is relaxed?

Question

Can such non - Euclidean geometries actually be imagined visually or can they be worked on only analytically like 4 dimensional worlds? If they can be imagined, and we relax Euclid's parallel postulate, what exactly do the previous postulates and definitions even mean? As apparently by relaxing the parallel postulate, we achieve geometries where the sum of the interior angles of a triangle isn't 180 degrees anymore but then the triangle itself in such worlds is curved and the angles are not sharp. Now if the angles themselves are not sharp and resemble a curve how do we give a certain degree or radian measure to it? And lastly does a geometry exist where a 'straight line' in the Euclidean sense is non existent, if so what restraint does that system have that says that such a thing is impossible. I mean what mathematical equation or axiom of the system disallows straight lines or sharp angles? Thank you very much for your time!

Answer 1

The shortest answer to your question refers to particular models: The spherical model of elliptic geometry, and the Poincaré disk model of hyperbolic geometry.

1. Define a point of the elliptic plane to be a pair of antipodal points on the unit sphere, a line between two points to be the great circle containing the two pairs of antipodal points, and the angle between two lines to be the ordinary Euclidean angle (in space) where two great circles meet.

2. Define the hyperbolic plane to be the open unit disk in the Euclidean plane, a point to be an element of this open disk, a line between two points to be the unique arc of a circle containing the two points and meeting the unit circle at a right angle, and the angle between two lines to be the ordinary Euclidean angle where two circle arcs meet.

In each case, it turns out that Euclid's first four postulates are satisfied. These are not the only models of elliptic and hyperbolic plane geometry, but it turns out that (aside from an overall choice of "unit length") any two models of elliptic geometry are abstractly equivalent, and any two models of hyperbolic geometry are abstractly equivalent.