Although I read some threads
hermitian distance functions and geometry in complex space
Geodesics and Distance in Hyperbolic Space
Distances in geodesic triangles
https://www.jstor.org/stable/27958625?seq=1
They are FAR, FAR above my ability to understand. I have done trig, linear algebra in multiple dimensions, calculus at the university level but never was exposed to differential geometry. I am basically a simpleton.
I cannot get a handle on how to calculate distance in a nonEuclidean Geometry. For example, we know that distance in euclidean space is calculated as the root of the sum of the squared distances in the x and y directions. Basically the hypoteneuse of a right triangle. In spherical space, with a vertex of the triangle at the pole and the opposite side at the equator, then the hypotenuse is equal to the vertical leg. I even figured out how to calculate distance on a paraboloid and ellipsoid, but these are simple geometric shapes.
How can I define a surface such that the the hypotenuse of a triangle is the sum of the length of the two legs?
This post
Why don't we have many non euclidean geometries out there?
seems to come close to what I am asking, but even it is at the limits of my grasp. Do I need to study differential geometry to figure this out? I viewed thi
https://www.youtube.com/playlist?list=PLBY4G2o7DhF38OEvEImfR2heX7Szmq5Gs
but it seems somewhat abstract.
I do not even know what I need to know.
