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This isn't a full answer, but I think that there is something wrong with how we are doing the axes for polar plot - probably because of our aspect ratio handling (?).

Compare

sage: polar_plot(1/sin(1.01*theta),(theta, pi/12,pi/2))

with

sage: polar_plot(1/sin(1.001*theta),(theta, pi/12,pi/2))

and you'll see what I mean.

This is now Ticket 11978.