Non-symmetrical lemniscate curve parameterization I'm trying to fit a function to data points. The data generally resembles a butterfly/lemniscate shape, see drawing.
The problem is that the shape in my data can be rotated, skewed and/or non-symmetrical.
I've been looking at Bernoulli's, Devil's curve, Watt's curve, however, these are, as far as I can see, symmetrical.
Does anyone know of a plane curve that is able to represent the example shapes? Preferably in Cartesian coordinates.

 A: Mathematical Models, H.M. Cundy and A.P. Rollet,  page 71:

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Interesting Quartics
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(8) $x^4 = x^2 - y^2$ (lemniscate)

Playing with geogebra, I think I found a nice generalization of the lemniscate that you can use. It's the following equation:
$$a(x- c_1)^4 -b(x - c_1)^2 + c(y - c_2)^2 + d(x - c_1)(y - c_2) + e(x - c_1)^3 = 0$$
Playing with the parameters, seemes like I can deform the lemniscate on the plane (including asymmetric deformations) at will. I will upload a GeoGebra app soon, so you can see it for yourself.
Surely, maybe including more terms would make it more general, and might capture a deformation that's missing. I've tried a few terms (like $y^3$), but any coefficient other then 0 would destroy the lemniscate shape. Also, this is not rigorous in any kind, it's purely empirical, so I would watch out before using it on any real application.
EDIT: Here's my GeoGebra file. Also, as a note, check that are bounds for some of the coefficients used, otherwise the lemniscate form is also lost.
A: Can you consider free form curves such as Bezier ?


You can also use a standard lemniscate and deform its plane, for instance by an homographic transform.

A: 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:
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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
