The polar equation to any curve can be written r=f(θ).

If P and Q are inverse points with respect to a circle whose centre O is at the pole and whose radius is k, then, since OP OQ = k2, it follows that the polar equation to the inverse curve is k2/r=f(θ).

It is proposed to find by inversion some properties of the limaçon and the cardioid that follow from certain well-known properties of the conies : others will suggest themselves. For convenience the same letter is used to denote both a point on the original figure and the corresponding point on the figure obtained by inversion.