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 - x ⁿ )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 ...