Shifting spiral of archimedes over to the right and down

My only advice is to convert to rectangular, shift, try to convert back to polar. Unfortunately, since you have just $\theta$ you'll have to use arcsine/arccosine and that's not good because it's only defined when the argument is in the appropriate interval. But if you do it you get $r = a + b\theta = \sqrt{x^{2} + y^{2}} = a + b \sin^{-1}\Big( \frac{x}{\sqrt{x^{2} + y^{2}}}\Big)$ . Shifting by $c,d >0$ , $\sqrt{(x-c)^{2} + (y+d)^{2}} = a + b \sin^{-1}\Big( \frac{x-c}{\sqrt{(x-c)^{2} + (y+d)^{2}}}\Big)$ and you can have fun trying to convert this back to just $r$ and $\theta$ .