where n, a and b are positive numbers.
This formula defines a closed curve contained in the rectangle −a ≤ x ≤ +a and −b ≤ y ≤ +b. The parameters a and b are called the semi-diameters of the curve.
When n is between 0 and 1, the superellipse looks like a four-armed star with concave (inwards-curved) sides. For n = 1/2, in particular, the sides are arcs of parabolas.
When n is 1 the curve is a diamond with corners (±a, 0) and (0, ±b). When n is between 1 and 2, it looks like a diamond with those same corners but with convex (outwards-curved) sides. The curvature increases without limit as one approaches the corners.
When n is 2, the curve is an ordinary ellipse (in particular, a circle if a = b). When n is greater than 2, it looks superficially like a rectangle with chamfered (rounded) corners. The curvature is zero at the points (±a, 0) and (0, ±b).
If n < 2 the figure is also called an hypoellipse; if n > 2, a hyperellipse.
When n ≥ 1 and a = b, the superellipse is the boundary of a ball of R2 in the n-norm.
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