Equation of an 'n-Oval' (pins and loop of string method)

Question

You can write the equations of an n-ellipse (polyellipse, egglipse, k-ellipse, etc...), where points $(x_1,y_1), (x_2, y_2)...,$ are foci:

$$\sqrt{(x-x_1)^2+(y-y_1)^2}+\sqrt{(x-x_2)^2+(y-y_2)^2}+\sqrt{(x-x_3)^2+(y-y_3)^2}+\sqrt{(x-x_4)^2+(y-y_4)^2}...=d$$

For two points, the curve can be the same as an ellipse drawn with two pins and a loop of string (see fig. 1).

However, with more points, the curve seems to be the same as James Clerk Maxwell's construction method (see fig. 2)

You can also create various ovals by adding points to the loop and pins method shown in the first figure (see also fig. 3).

What I want to know is how to construct equations for curves of the kind shown in fig. 3 given any number of points (akin to the above equation). I'd also be interested in knowing if there's anything published on this type of curve. Thanks.

Answer 1

The figure below is a reproduction of your fig. 3: $ABCDEF$ are the pins and the string is held tight by the pencil at $P$.

As long as $P$ lies between the extensions of lines $FA$ and $BA$ we have that $PF+PB$ is constant. Hence $P$ describes an arc $HI$ of the ellipse with foci $F$ and $B$ (blue in the figure).

After $I$, point $P$ lies between the extensions of $FA$ and $CB$, with $PA+PB$ a constant. In this case $P$ describes an arc $IJ$ of the ellipse with foci $A$ and $B$ (orange in the figure). Note that the ellipses are tangent at $I$, because they share the same normal (the bisector of $\angle FIB=\angle AIB$).

After $J$, point $P$ lies between the extensions of $FA$ and $DC$, with $PA+PC$ a constant. In this case $P$ describes an arc of the ellipse with foci $A$ and $C$ (not shown in the figure), and so on.

In summary, the curve is a patch of 12 elliptical arcs, each arc being tangent with the preceding and following one.

Answer 2

An idea.

I agree of course with the answer of Intelligenci Pauca, but I would like to say that there is some hope to have a single equation using absolute values like in the examples below (for different values of $p$) ; these curves are made of connected conical arcs (some of them may not be elliptical arcs) ; moreover, in the points where the arcs are connected, there is no smoothness. But I am almost certain that it is possible, at least in certain cases, to assemble second degree polynomials with or without absolute value(s) in order to get the curves you want.

$$3x^2+2y^2+|y+2x|+|xy|=p$$

SAGE program for the generation of the figure :

 var('x y');
 g=Graphics();
 for k in range(0,20) :
    p=k/8
    g+=implicit_plot(3*x^2+2*y^2+abs(y+2*x)+abs(x*y)==p,(x,-1,1),(y,-1,1))
 show(g)