# What type of curve is described by $\cos{x}+\cos{x}\cos{y}+\cos{y}=0$

curvesimplicit functionplane-curves

Does the curve by the function $$\cos{x}+\cos{x}\cos{y}+\cos{y}=0\\x=[-2\pi/3,2\pi/3],\; y=[-2\pi/3,2\pi/3]$$ belong to any known curve family? Examples of curves can be found in Wikipedia (Link1, Link2) but the longest list with $$\approx 1000$$ curves you find here.

This implicit equation can be written

$$(1+\cos x)(1+\cos y)=1$$

Using a classical trigonometric formula:

$$4(\cos(x/2)\cos(y/2))^2=1$$

$$2\cos(x/2)\cos(y/2)=\pm 1$$

Due to invariance with respect to changes $$x \to -x, y \to -y$$ (in connection with symmetries with respect to coordinate axes), one can reduce the study to the first quadrant with equation:

$$\cos(y/2)=\frac{1}{2 \cos(x/2)}$$

finally giving a cartesian equation for the curve in the first quadrant:

$$y=2 \arccos \left(\frac{1}{2 \cos(x/2)}\right)$$

This form doesn't evoke more anything known than the implicit form (but see the Edit below).

A first check : if $$x=0$$ we get $$y=2 \frac{\pi}{3}$$ which is the point $$(0,2 \frac{\pi}{3}\approx 2.09)$$ on your curve.

Remark: there is also another symmetry with respect to line bissector $$y=x$$.

Important edit: With the help of Geogebra, I have found a very good fit of this curve with the following so-called "squircle" with equation

$$|x|^{2.42}+|y|^{2.42}=5.95\tag{1}$$

as we can see on the following representation (the red curve with equation (1) hides almost completely the initial curve, in black). It would be interesting to understand why such a good fit exists.