Non-symmetrical lemniscate curve parameterization
There is a polar asymmetric modelling for the lemniscate: 



which you can rotate and scale more easily that in Cartesian coordinates (but easy to convert). Such parameterizations are used very often as they can be less troublesome to fit.

Similar curves in astronomy are also called analemmas:

.

From an image processing fit point of view, you can consult: A Unified Scheme for Detecting Fundamental Curves in Binary Edge Images, Asano, Tetsuo et al., Computational Geometry, 2001