gregeganSF, 9 months ago

A sphere is the nicest surface with constant Gaussian curvature κ=1, but these surfaces also have κ=1, except at the cusps.

In cylindrical coords:

z = √(1-χ²) E(arcsin(ρ/χ) | χ²/(χ²-1))

z is measured from a cusp
χ is radius of equator
E is an elliptic integral of the 2nd kind

A sphere deforms, stretching along one axis and shrinking in width into something like a prolate ellipsoid, but with a cusp at the poles.