I discovered this curve that looks like a lemniscate. It’s defined as the set of points $P$ in ${\Bbb R^2}$ that satisfies the equation $$\frac{1}{d_{-1}(P)}+\frac{1}{d_{1}(P)}=2,$$where $d_{a}(P)$ is the distance from $P$ to the point $(a,0)$. It looks like this. My two questions are:
- Has this curve been studied before / does it have a name?
- Could someone point out a method for (if possible) finding an algebraic expression for the length or area enclosed by this curve?
There a few things I have noticed; the curve intersects the $x$-axis at $x=0$ and $x=\pm\varphi$ (where $\varphi$ is the golden ratio), and it can be parameterized in the upper half plane by $$x(t)=\frac{t^4-t^3}{(2t-1)^2}$$$$y(t)=\sqrt{t^2-(x(t)+1)^2}$$ for $\varphi-1\leq t\leq\varphi+1$.
Describing the distances using the Pythagorean theorem and trying to solve for $x$ or $y$ doesn’t seem to get me anywhere since it results in something like an eight-degree equation.
The area and length of the curve are pretty close to whole numbers: $$A=3.09404630588...$$$$l= 8.99599858883...$$