Finding the perimeter of the shape of $x^2+x^2y^2+y^2=a^2$

Question

I need to find the perimeter of the shape of $$x^2+x^2y^2+y^2=a^2$$ ($a$ is any real number) in the Cartesian plane.

Is there any general formula that depicts the ratio of area to perimeter of every given shape?

Answer 1

Not a full answer, but a help...

First of all, trying to find the ratio of area to perimeter is like multiplying difficulties. In particular, this ratio isn't a constant in general.

Here is how I have considered the issue.

I have begun by a plot of the curves for increasing values of parameter $a$ (see figure below, the largest curve being for $a=4$).

Fig. 1 : The curves are convex till $a=2$, non-convex beyond.

The equation is given in implicit form $f(x,y)=0$ ; on this equation, we see different things : $$\underbrace{x^2+y^2}_{\text{dominant for small}\ a}+\underbrace{x^2y^2}_{\text{dominant for large}\ a}=a^2$$

for example, the fact that the curves are like circles for small $a$ and like hyperbolas for large values of $a$; the fact that they are invariant for different symmetries ($x \leftrightarrow -x, y \leftrightarrow -y, x \leftrightarrow y$) that we evidently find back on the curves. Consequence : it suffices to study theses curves in the first quadrant (the first half of the first quadrant would even be enough).

We can switch to an equivalent cartesian equation (exercise!),

$$y:=\pm f_a(x)=\pm \sqrt{\dfrac{a^2-x^2}{1+x^2}}$$

well, a pair of cartesian equations.

From here, you can compute the length of $1/4$ of the curve by a classical formula :

$$L_a=\int_0^{a} \sqrt{1+f'_a(x)^2}dx=\int_0^{a}\sqrt{1+\dfrac{x^2(a^2 + 1)^2}{(x^2 + 1)^3(a^2 - x^2)}}dx$$

It seems one cannot obtain an exact expression for $L_a=L(a)$. Numerical computations give the following curve, with good approximation by

$$M(a)=a \dfrac{6a-1}{\pi a+1}$$

(thanks to @Blue who has indicated that my first approximation wasn't exact).

Fig. 2 : Evolution of perimeter $L=L_a$ (Blue) as a function of $a$ and its approximation $M$ (Red). Interpretation : for $a=500$, one finds $L(a)=962.2$ giving a total perimeter $4 \times 962.2$

Answer 2

One can start from $$ y= f(x)=\sqrt{\frac{a^2-x^2}{1+x^2}}$$

A quarter of the star parametrized on a single $u, (0<u< \pi/2),\,(a,1,2,0.5)$ per Blue's comment:

$$ \tan \alpha= \sqrt{1+a^2},\, (x,y)= ({\tan \alpha \tan u -1},{\tan\alpha \cot u -1}\,) $$

with polar radius

$$ r^2 = 2 ( \frac{\tan \alpha}{\sin 2u}-1) $$

and polar angle $\theta$ (please check)

$$ \tan^2\theta=\frac{\sin(u-\alpha)}{\cos(u+\alpha)}. $$