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
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
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