Squircles

 The equation of the unit circle (that is, a circle of radius 1 centered at the origin) is famously given by 

Circles are just a special case of a far more general family of curves: squircles. These are given by the equation

For n an integer greater than 0 (n = 0 gives the empty set). We shall call this an n-squircle. Note that for any n > 1 these are smooth, bounded curves (the 1-squircle has sharp corners at (±1, 0) and (0, ±1)). This is also why we use the absolute value signs. Since, for odd n the equation xⁿ + yⁿ = 1 gives an unbounded curve. 

This curve is just the graph for (1 - xⁿ)¹/ⁿ, so it is not very interesting.

What happens as n → ∞?

Squircles thank their name to the fact that they look like something inbetween a circle and a square. For example, the 4-squircle looks as follows:

In fact, as n → ∞ squircles begin to look more and more like squares. Suppose we take some fixed x and take yₙ > 0 so that (x, yₙ) lies on the n-squircle. Then we have yₙ = (1 - xⁿ)¹/ⁿ. We can compute this limit as follows:

Since 1 - xⁿ → 1 so that ln(1 - xⁿ) → 0 and, of course, 1/n → 0. In other words, yₙ → 1 so that the point (x, yₙ) approaches (x, 1). Similarly, if we pick yₙ < 0 the point will approach (x, -1). 

Similarly, if we fix y and take xₙ > 0 so that (xₙ, y) lies on the n-squircle, then xₙ → 1 and if xₙ < 0 we get xₙ → -1. In other words, (xₙ, y) → (0, ±1). We can also see this visually:

Area and circumference

Let's compute the area of a squircle. Since it is symmetric with respect to the x- and y-axes, it actually suffices to compute area for x, y > 0 and multiply this by 4. In other words, if we denote the area of an n-squircle by Aₙ we get

In the above B(z, w) refers to the beta function and Γ refers to the gamma function. As n → ∞ we do indeed get that Aₙ → 4, which is the area of the square with corners (±1, 1) and (±1, -1). Furthermore, the area increases as n gets larger.

Another value we might be interested in computing is the circumference. This too can be achieved using an integral, though as far as I'm awara these do not have a "closed form" like the integrals for the area.

I would be quite surprised if it could be expressed. If we are being honest, even the area is kind of a  cheat since it used the gamma function. Also here we will have Cₙ → 8 since the n-squircle approaches the square smoothly.

Curvature

We consider the parametrisation for 0 < t < 1 by t ↦ (t, (1 - |t|ⁿ)¹/ⁿ. This then gives

This then allows us to compute the curvature via the following nightmare of a formula:

Which, after a lot of calculations, gives

assuming I made no errors. Note, in particular, that κ(0) = κ(±1) = 0 for n > 2. In other words, the points of no curvature are (0, ±1) and (±1, 0).