How do you offset a circle from the origin in polar coordinates or a sphere from the origin in spherical?
In other words, is there any way to restate a general rectangular formula for a circle:
In polar coordinates? How about:
(x-a)^2 + (y-b)^2 = c^2
(x-a)^2 + (y-b)^2 + (z-c)^2 = d^2
I ran across this concept as part of a Line Integral Problem on a take-home exam, where the field and path were both rectangular, but the path was a unit circle centered at (2,3). At first, I wanted to convert both to polar to make the limits of integration simpler. However, I wasn't able to figure out how to convert a rectangular transformation to polar coordinates. I ended up substituting x=x+2 and y=y+3 across the board (translating the circular path to the origin) and then converting to polar. However, I've already taken the final exam, and the professor still hasn't finished grading those take-home tests. I have no idea if my substitution worked out , and I haven't yet figured out how to do a rectangular translation in polar coordinates.
So... can anyone help me out before the finger I'm using to scratch my head wears down to a nub?