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Simple closed curves on a plane sorted by "complexity": circle, ellipse, what's next? Simple means the curve does not cross itself.

I am not able to define "complexity" in a rigorous way, let's say the more parameters are needed to define a curve, the more complex it is.

To me, the simplest simple closed curve is the circle because it is described by just three real numbers: the radius, and the $(x_C,y_C)$) coordinates of its center $C$.

The following is the ellipse described by five real numbers: the minor axis, the major axis, the center and the angle formed by the major axis and the horizontal axis of a Cartesian coordinate system.

What's next?

Are there any rigorous ways to sort curves?

Edit I am interested only in smooth curves.

Alessandro Jacopson
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    Polygons can be simple or complex, not sure how they will fit in this classification. – Vasili Mar 02 '23 at 15:50
  • @Vasili you're right! I am interested only in smooth curves. I am afraid my question is too fuzzy... – Alessandro Jacopson Mar 02 '23 at 16:00
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    Incidentally, my spouse was studying polar curves the last month or so. You may want to look up classification of polar curves. – Vasili Mar 02 '23 at 16:07
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    Could it be that you're just asking for the order of the curve? All of these curves can be represented as level sets of polynomials, in this case of degree 2. Solving for them locally gives you a graph of a polynomial of the second degree, which is uniquely determined by 3 points (you can think of them as parameters or some combination of them). The next kind could be level sets of polynomials of order 3 and so on? Of course some of them have certain symmetries which allow you to reduce the number of necessary parameters. – Boxonix Mar 02 '23 at 17:11

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One of many possible approaches.

Parametrize the curve $P$ (using complex notation) by specifying that the unit tangent vector $T$ pointing in the direction $T=e^{i \theta}$ has associated real-valued radius of curvature $R(\theta)$. Then $ dP = \frac{ dP}{ds} ds = T ds= e^{i \theta} \frac {ds}{d\theta} d\theta = e^{i \theta} R(\theta) d\theta$ can be integrated to reconstruct $P$. The radius of curvature function $R= \frac{ds}{d\theta}$ is periodic with period $2\pi$ (or if you wish to model exotic cases, a whole number of multiple thereof). The curvature function must therefore possess a periodic Fourier expansion, and can be classified crudely by the number of terms that arise in the Fourier expansion $R(\theta)= \sum_{k} c_k e^{i k \theta} + c_{-1} \theta$. Note that the condition that $R(\theta)$ be always real-valued imposes the symmetry constraint $\overline {c_{-k}}= c_k$ on the complex Fourier coefficients of $R(\theta)$.

Then reconstruction of $P$ involves only term wise integration of the associated Fourier series for $ e^{i\theta} R(\theta)= \sum_k c_k e^{ (k+1) \theta}$, which is easy, giving $P= c_{-1} \theta +\sum_{k \ne -1}c_k \frac{1}{ i(k+1)} e^{i (k+1) \theta} $ modulo a constant of integration that translates the curve in the plane. Note that if $c_{-1} \ne 0$ then integration produces a term in $P(\theta)$ that is linear in $\theta$. This generates a non-closed curve that wraps in spiral fashion about itself with equal increments with each revolution. So generally speaking $k=\pm 1$ is a forbidden pair of frequencies in this model.

Classification. The circle requires only $R=$ constant (the null frequency $k=0$). The ellipse occurs when you have only frequencies $k=0,2,-2$.

You can model non-convex curves (e.g kidney-shapes) by allowing $R(\theta)$ to change sign.

In general you can play with various sophisticated measures of the complexity of the curve by analyzing the pattern of the relative magnitudes of the Fourier coefficients (the spectral profile of $R(\theta)$.

The tricky part/drawback of this method is avoiding self-crossings of the curve, which can sometimes occur when there are large sign changes in $R(\theta)$.

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