Is it possible to change the pole and/or the polar axis in a polar coordinate system?

Citing Wikipedia's article on polar coordinates...

In mathematics, the polar coordinate system is a two-dimensional coordinate system in which each point on a plane is determined by a distance from a fixed point [pole] and an angle from a fixed direction [polar axis].

$x = r\cos{φ}$

$y = r\sin{φ}$

$r = \sqrt{x^2 + y^2}$

$φ = \arctan2{(y, x)}$

The article expounds further in the below image, with $O$ being the pole and $L$ being the polar axis. Is it possible to change the coordinate system's pole and/or polar axis (e.g., setting $-L$ as the polar axis)?

Yes this is possible, and is in fact done all the time. If you want the pole to be at $(x_0,y_0)$ you just make the following substitutions in your equations.

$$x \rightarrow x-x_0$$ $$y \rightarrow y-y_0$$

Edit: To answer the question about changing the polar-axis.

One way this can be done is by rotating the coordinates so that the $x-axis$ is at an angle to what it used to be. To rotate the coordinates by and angle $\theta$ apply the transformation,

$$x' = x \cos(\theta) - y \sin(\theta)$$ $$y' = x \sin(\theta) + y \sin(\theta)$$

In that case what you would want to substitute into the polar coordinate equations is,

$$x\rightarrow (x-x_0) \cos(\theta) - (y-y_0) \sin(\theta)$$ $$y \rightarrow (x-x_0) \sin(\theta) + (y-y_0) \sin(\theta)$$

• Thanks, @Spencer. I suspected that; but what about changing the polar axis? – Noob Saibot Feb 10 '14 at 21:41
• Fair warning: My signs may be a bit off in the coordinate rotation. If the polar axis rotates in the wrong directions just change $\theta \rightarrow -\theta$. – Spencer Feb 10 '14 at 21:51

It is indeed possible to change the pole by using the Law of Cosines for Spherical Triangles. One practical application would be if you model Earth as a perfect sphere and you want to magnetic latitude of a given point.

Let's say you know your latitude and longitude, you know the latitude and longitude of the Earth's geomagnetic pole, and you want to find your magnetic latitude. The Law of Cosines for Spherical Triangles will yield the cosine of the latitude like so: $$\cos c=\cos(a)\cdot\cos(b)+\sin(a)\cdot\sin(b)\cdot\cos(C)$$ where:

• $$c$$ is your magnetic latitude,
• $$a$$ is the geographic latitude of the geomagnetic north pole,
• $$b$$ is your geographic latitude,
• $$C$$ is the difference between the longitude of the geomagnetic north pole and the longitude of your position.

Note that to do a complete coordinate transformation, it's not enough to simply declare a new pole and find your latitude from that pole; you'll also have to have a new rotational reference coordinate (in geography, a new prime meridian), so this only works for getting your new geomagnetic latitude; there is no geomagnetic longitude.