New Project Looking for Help: Plotting great circles

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Hi dividenot,

... I must say though, I'm somewhat confused about the ask/answer system here. ... Is editing the message like this the best way to have a discussion?

We're sort of figuring out how this site can be useful as we go -- it's not that old yet. Only people with sufficiently hight karma can edit the wiki question, so in general it's not the best way to go; I did it in this case because my comments didn't really seem like an "answer".

Initially I was looking for some kind of forum where I could have a discussion or look up any needed information but ask.sagemath.org was the closest thing I could find.

There are also the sage-* google groups -- the two most relevant are probably sage-support and sage-devel. They're both very active, and the only down side is that you have to be approved to join the groups; that usually takes less than a day though.

Now on to something more like an answer: Looking at the documentation for 3d graphics, I found that Sage has support for alternate (non-cartesian) coordinate systems; e.g. spherical coordinates -- no rotation matrices needed. To me, the documentation was a little confusing (especially the part where, I think, the words dependent and independent are interchanged!), but eventually I realized that you can plot a "segment" of a sphere simply with:

sage: T = Spherical('radius', ['azimuth', 'inclination'])
sage: var('theta, phi')
sage: plot3d(2, (theta, 0, pi/4), (phi, 0, pi), transformation=T, aspect_ratio=(1,1,1))

Now comes the real fun: you can make a very narrow section with, e.g.

sage: plot3d(2, (theta, 0,.01), (phi, 0, 2*pi), transformation=T, aspect_ratio=(1,1,1))

and, even better, define a function which returns the plot starting at theta=t:

sage: cp = lambda t: plot3d(2, (theta, t, t+.01), (phi, 0, 2*pi), transformation=T, aspect_ratio=(1,1,1))

You can draw multiple plots in the same picture by adding, or, if you have a lot of things to add, use summation over a range!

sage: many_circles = cp(0) + sum(cp(pi/(2^n)) for n in range(1,7)) 
sage: many_circles.show()

I don't know if you can do it using plot3d via a spherical transform, but you can just do a 3 dimensional parametric plot directly providing the transformation as follows:

var('phi')
r = 1
theta = 0
parametric_plot3d((r*cos(theta)*sin(phi), r*sin(theta)*sin(phi), r*cos(phi)), (phi, 0, 2*pi))

Then as above, you can create a function and show many circles

cp = lambda theta: parametric_plot3d((r*cos(theta)*sin(phi), r*sin(theta)*sin(phi), r*cos(phi)), (phi, 0, 2*pi))
many_circles = cp(0) + sum(cp(pi/(2^n)) for n in range(1,7)) + sum(cp(-pi/(2^n)) for n in range(1,7))
many_circles.show()