How can one transform polar functions along rectangular coordinates?

Question

Suppose one has a function $r = g(\theta).$ What I would like to do is control how it shifts horizontally and vertically as I would in a rectangular function $ y =f(x+a)+b.$ Is there a reliable way to do this?

Answer 1

There is not a nice formula, but one can compute the new $r,\,\theta$ using the laws of cosine and sine.

In the case of the shift of a polar point $A=(r_1,\theta_1)$ to the right by an amount $h$ to the point $A'=(r_2,\theta_2)$ we have by the law of cosines that

$$ r_2^2=r_1^2+h^2-2r_1h\cos(\pi-\theta_1) $$

and by the law of sines that

$$ \sin(\theta_2)=\frac{r_1\sin(\pi-\theta_1)}{r_2} $$

One could analyze left, up and down shifts in a similar fashion.