[Math] How to translate a circle to new centre

circlestransformation

Consider that we have a circle drawn a round the origin (0,0). That circle has some points drawn on its circumference. Each of those points has range and azimuth $(r,\theta)$, the r and $\theta$ values of these points are calculate with responding to the origin (0,0).

I want to move or translate that circle and its (on-circumference) points to a new center (x,y), the new center can be in any quadrant. Here is an image for more demonstration (consider that it was translated to the first quadrant):enter image description here

The question is what is the new $(r, \theta)$ of each point with responding to the origin (0,0) after translation to the new center?

Best Answer

For a circle centered at the origin we have (taking $x$ axis on the right and $y$ axis upward and $\theta$ as indicated in the sketch):

$$x^2+y^2=R^2 \iff (x,y)=(R\sin \theta_0,R\cos\theta_0)$$

for a circle centered at $C=(x_C,y_C)$ we have:

$$(x-x_C)^2+(y-y_C)^2=R^2\iff (x,y)=(x_C+R\sin \theta_0,y_C+R\cos\theta_0)$$

Then the new polar coordinates are (using the arctan2 function) are :

  • $r=\sqrt{(x_C+R\sin \theta_0)^2+(y_C+R\cos\theta_0)^2}$
  • $\theta=\arctan2\left(\frac x y\right)=\arctan2\left(\frac {x_C+R\sin \theta_0}{y_C+R\cos\theta_0}\right)$
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