Name of the family of curves

Question

I tried to generalize the Lemniscate of Gerono.

Instead of $x = \dfrac{t^{2} - 1}{t^{2}+1}$ and $y = \dfrac{2t(t^{2}-1)}{(t^{2} - 1)^{2}}$, I used $x = \dfrac{a(t^{2} - 1)}{t^{2}+1}$ and $y = \dfrac{bt(t^{2}-1)}{(t^{2} - 1)^{2}}$.

When eliminating $t$, I used the positive root instead of both. What I got is this:

$y = \dfrac{bx}{a}\left(\dfrac{a - x}{2a}\right)\sqrt{\dfrac{a + x}{a - x}}$

Out of curiosity, I replaced $y$ by $y^{2}$. Can I ask what is the name of the family of the curve $y^{2} = \dfrac{bx}{a}\left(\dfrac{a - x}{2a}\right)\sqrt{\dfrac{a + x}{a - x}}$?

Answer 1

In fact the equation of your curve can be simplified into:

$y^{2} = \dfrac{bx}{2a^2}\sqrt{(a + x)(a - x)}\tag{1}$

with the very tiny difference that here $x \in [0,a]$ whereas in the initial version, $x \in [0,a)$.

This type of curves belong to the general family of quartics like other lemniscates because when both sides of (1) are squared, equation:

$y^{4} = \dfrac{b^2x^2}{4a^4}(a^2 - x^2)\tag{2}$

is the zero set of a 4th degree polynomial.

The curve with equation (1) hasn't the shape of an $\infty$, therefore cannot be called a generalized lemniscate, because $x$ cannot take negative values.

Whereas, with equation (2), you would get a "bow-tie" lemniscate (the same curve with its symmetrical part with respect to the ordinate axis)

Fig. 1: The "bow-tie" lemniscate. In blue: curve with equation (1) (case $a=5, \ b=3$). In red+blue: curve with equation (2).

Besides, here are two pieces of information that can have their interest in your attempt to find other types of curves with the $\infty$ shape:

1. Are you aware that the lemniscate of Gerono can be considered as the projection of Viviani's curve (see corresponding Wikipedia article), which is the intersection of a sphere centered in $O$ with radius $OA=R$ and a cylinder with diameter $OA$ ? Therefore your research could be oriented towards a definition taking into account a generalization of the sphere and/or a generalization of the cylinder.

2. In a recent answer of mine here, I have had the opportunity to work on a $\pi/4$ rotated version of Gerono's lemniscate. It could be beneficial to your extension to attempt this kind of transformation.

Web references: Lemniscate of Gerono - Mathcurve and this one showing the connection with Bernoulli's lemniscate, somewhat the "archetype" of all lemniscates.