Superellipses and Superellipsoids

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Back in the 1970's, I came across the aphoristic poems of Piet Hein. Called grooks in English, they were brief, terse and immensely wise. Being in the prime of my age for invention, and minding mathematics more than at any time since, I was also intrigued by his advocacy of the superellipse for the solution of practical problems where contrasting requirements had to be reconciled. He didn't invent them, but he breathed life into them.

An ordinary ellipse is, as any fule kno, a generalisation of the circle, having the equation

The points (a,0) and (0,b) lie on the perimeter and the line segments from the origin to these points are called the semi-axes, of length a and b respectively, major or minor depending on their relative magnitude.

The superellipse simply generalises the exponent 2 to any larger (hyperellipse) or smaller (hypoellipse) positive value n. To avoid headaches outside the principal quadrant with odd powers of n, absolute values of x and y are used:

abs(x/a)ⁿ  +  abs(y/b)ⁿ  =  1

The results are fascinating, as you can see by clicking here. If the outer dome of St Paul's Cathedral really is slightly flattened at the top (rather than being visually truncated by the lantern), then an appropriate superellipse would surely have appealed to our dynamic duo, Hooke and Wren, as generatrix of the corresponding superellipsoid.