Given $n$ distinct points $\mathbf{x} = (\mathbf{x}_1, \ldots, \mathbf{x}_n)$ in the plane $\mathbb{R}^2$, I associate a real analytic map:

$f_{\mathbf{x}}: S^1 \to \mathbb{R}^2$

with the following properties. If $C_n(\mathbb{R^2})$ is the configuration space of $n$ distinct points in $\mathbb{R}^2$, then each there is a real analytic map $$f: C_n(\mathbb{R}^2) \times S^1 \to \mathbb{R}^2$$ such that $f_{\mathbf{x}}(-) = f(\mathbf{x},-).$$

The image of $f_{\mathbf{x}}$ is in the convex hull of $\mathbf{x}$. Also, if $\mathbf{x}'$ is a permutation of the points in $\mathbf{x}$, then $f_{\mathbf{x}'} = f_{\mathbf{x}}$. Finally, if you apply a Euclidean transformation to the configuration $\mathbf{x}$, then the image of the curve $f_{\mathbf{x}}$ gets transformed by the same Euclidean transformation. Here are some plots.

The points in $\mathbf{x}$ are shown in red, and the corresponding curve is shown in blue.

My question is, is there any interest in such Bezier-like curves? Can I publish them somewhere? I am not familiar with the area. Could someone possibly suggest some journal(s) by any chance?

Actually, I found originally a related family, where given $n$ distinct points $\mathbf{x} = (\mathbf{x}_1,\ldots,\mathbf{x}_n)$ in $\mathbb{R}^3$, I associate a real analytic map:

$$f_{\mathbf{x}}: S^2 \to \mathbb{R}^3,$$

with similar properties as the first family of maps. My work does not extend further to higher dimension though. Any comments and/or suggestions are welcome.

Edit 1: after discussing with @DanieleTampieri, and looking at the third figure, it does seems like my maps could possibly be used in boundary detection problems possibly. It is interesting that a single formula seems to accomplish what is usually done algorithmically.

Edit 2: following one of @JochenGlueck's comments, I did the following experiment. I started with a configuration of $4$ points and plotted the corresponding curve. Then I added $6$ points at random inside the convex hull of the initial $4$ points, and plotted the corresponding curve. The new curve looks like it passes through the original $4$ points now, interestingly. Here are the corresponding two plots.

Edit 3: I wrote a GUI interface using the Python library Tkinter. Now I can place the points on a canvas, press a button and see the resulting curve. This will enable me to experiment further with these curves. After consulting with a few people, I think this may fit in a journal of Computer Graphics perhaps. It may not be appropriate in a journal of Approximation theory or Numerical Analysis I think, as it does not really contain results, as of now. In any case, I will let things stew a bit more in my brain before writing things up. Thank you all.

Edit 4: I saw how to generalize my maps in two different directions: as a sequence of maps, and in higher dimension. More specifically, I have defined, for each positive integer $m$, a real analytic map: $$f_m: C_n(\mathbb{R}^d) \times S^{d-1} \to \mathbb{R}^d,$$ such that, given $\mathbf{x} \in C_n(\mathbb{R}^d)$ (where $C_n(\mathbb{R}^d)$ denotes the configuration space of $n$ distinct points in $\mathbb{R}^d$), the map $f_m(\mathbf{x},-)$ maps the $d-1$ dimensional sphere to a good approximation of the boundary of the convex hull of $\mathbf{x}$. In the cases I considered, it seems that small values of $m$ suffice in practice (I mostly experimented in 2d and a little bit in 3d). Here is the plot of the images of some sample points on the sphere, for the case where $\mathbf{x}$ is the configuration of the $4$ vertices of a regular tetrahedron. I used $m=3$. I also include a 2d example, with $m=3$ (my previous map corresponds to $m=1$).

Edit 5: I uploaded a short note on arXiv. In case someone is interested in knowing how my maps are defined: *Rational Maps and Boundaries of Convex Hulls*, arXiv:2004.04538. I submitted this short note to a journal for review (sorry for all the updates).

Edit 6, UPDATE: Peter Olver found a counterexample to my conjecture in the arXiv article in Edit 5, so I will withdraw it, BUT this led to a fruitful collaboration where, after modifying the definition of the maps, we were actually able to *prove* convergence in the preprint *Continuous Maps from Spheres Converging to Boundaries of Convex Hulls*, arXiv:2007.03011. However, the maps are only piecewise rational, yet they are continuous. I am curious to know what people think of our preprint, for the ones who are interested enough to read it!

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