Non-symmetrical lemniscate curve parameterization

Question

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.

Answer 1

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

$\vdots$

Interesting Quartics

$\vdots$

(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.

Answer 2

Can you consider free form curves such as Bezier ?

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