So, i'm working my way through "Geometry from a Differentiable viewpoint" (or, trying to get through section 1.1, anyways).(adsbygoogle = window.adsbygoogle || []).push({});

right now, it's spherical geometry. so far, a great circle has been defined as the set of points on the sphere that intersect with a plane that intersects the origin of the sphere. The area of a lune, the "sherical pythagorean theorem", and the spherical sine theorem have all been presented with very brief proofs. It has taken me 5 hours of good work to get through 5 pages of this stuff.

the exercise that I'm doing is this : "The sphere of radius one can be coordinatized as teh set of points [tex](x,y,z)[/tex] in [tex]\mathbb{R}^3[/tex] satisfying [tex]x^2 + y^2 + z^2 = 1 [/tex], or as the set of points [tex](1,\psi,\theta)[/tex] in spherical coordinates, with [tex]0 \leq \psi \leq 2\pi [/tex], and [tex]0 \leq \theta \leq \pi [/tex]. In these two coordinate systems, determine the distance along great circles between two arbitrary points on the sphere as a function of the coordinates."

I worked on this thing for quite a while, first in spherical coordinates (they make more sense), then, deciding maybe i should use the [tex]\mathbb{R}^3[/tex] coordinate grid. i got essentially nowhere. finally, i looked at the answer, hoping to see something I missed. I did, kind of, but after working on those, i still don't have any idea how they got the answers, and I feel like i'm missing something essential to doing this stuff.

the Answers are : for [tex]\mathbb{R}^3[/tex], the great-circle distance between A and B (the arbitrary points) = [tex]\arccos(A \cdot B)[/tex], and for spherical coordinates is [tex]\arccos (\cos (\psi_1 - \psi_2)\sin \theta_1 \sin \theta_2 + \cos \theta_1 \cos \theta_2)[/tex]

So, if anyone can tell me maybe what I could read to get myself filled in on this, that would be great. I haven't looked at this in a couple days, so, I'm going to go ahead and continue working on it, and checking up on this, as this book looks super cool, but I want to make sure I have the proper foundations. By the way, i have taken and aced calc I and II, and matrix theory/linear algebra, so that you know what kind of background I have.

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# Spherical geometry, some simple things

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