User:Pepkin88/sandbox

$$a = \sqrt{\frac{R - r}{R + r}}$$

$$b = \frac{1 - a}{1 + a}$$

$$f(\phi) = \phi + 2 \arctan \frac{b \sin \phi}{1 - b \cos \phi}$$

$$f(\phi) = \phi + 2 \arctan \frac{\frac{1 - \sqrt{\frac{R - r}{R + r}}}{1 + \sqrt{\frac{R - r}{R + r}}} \sin \phi}{1 - \frac{1 - \sqrt{\frac{R - r}{R + r}}}{1 + \sqrt{\frac{R - r}{R + r}}} \cos \phi}$$