Inversion of conics

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The following cardoids are obtained by inverting conics of Newtonian polar form $ (r\rightarrow a^2/r), a=1 $, about a focus of variable eccentricity $e=(0.5,0.8,5/4,4/3,3/2). $

$$ \dfrac{a}{r}=\dfrac{1-e \cos \theta}{1-e^2}$$

Can the $e$ be interpreted geometrically on the cardoids without reference to the un-inverted conic?

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