Astroidal Ellipsoid

Astroidal Ellipsoid is a surface, given by a parametrisation, immersed.

aperiodic def-parametric immersed implemented misc

Formula

x(u, v)

\left(\cos\left(u\right) \cos\left(v\right)\right)^{3}

y(u, v)

1.5 \left(\sin\left(u\right) \cos\left(v\right)\right)^{3}

z(u, v)

2 \sin\left(v\right)^{3}

semi-axes (1, 1.5, 2) fixed to the operator defaults; equivalently the implicit (x/a)^(2/3) + (y/b)^(2/3) + (z/c)^(2/3) = 1

Chart reproduced numerically against the shipped implementation: fwd max 0.79 of tol, 0.0% of oracle beyond tol (tol floor 2.5 x 0.007876).

D. von Seggern, "CRC Standard Curves and Surfaces" (1993), p. 304 -- Fresnel's elasticity surface.
A. Robin (2004), plotted after M. Trott, "The Mathematica GuideBook for Graphics" (2004), p. 103 -- the paper bag surface.
O. Knill (2017); M. A. Villarino and J. C. Varilly (2024) -- the trihyperboloid and its volume ln 256.
C. Dupin, "Applications de Geometrie et de Mechanique", Paris 1822;
B. Odehnal, "Ortho-Circles of Dupin Cyclides", J. Geometry Graphics 10 (2006) 1-21 -- the inversion construction and the ring / horn / spindle classification.
J. Tannery, Bulletin des sciences mathematiques, 2e serie, 16 (1892) 190; O. Zoll, "Ueber Flaechen mit Scharen geschlossener geodaetischer Linien", Math. Ann. 57 (1903) 108-133; A. L. Besse, "Manifolds all of whose Geodesics are Closed", Springer 1978.