Fresnel Elasticity Surface

Fresnel Elasticity Surface is a surface, given by a parametrisation, immersed.

aperiodic def-parametric immersed implemented misc

Formula

x(u, v)

\sqrt{\sin\left(u\right)^{2} \cos\left(v\right)^{2} + 2.25 \sin\left(u\right)^{2} \sin\left(v\right)^{2} + 4 \cos\left(u\right)^{2}} \sin\left(u\right) \cos\left(v\right)

y(u, v)

\sqrt{\sin\left(u\right)^{2} \cos\left(v\right)^{2} + 2.25 \sin\left(u\right)^{2} \sin\left(v\right)^{2} + 4 \cos\left(u\right)^{2}} \sin\left(u\right) \sin\left(v\right)

z(u, v)

\sqrt{\sin\left(u\right)^{2} \cos\left(v\right)^{2} + 2.25 \sin\left(u\right)^{2} \sin\left(v\right)^{2} + 4 \cos\left(u\right)^{2}} \cos\left(u\right)

the radial quartic r = sqrt(a^2 l^2 + b^2 m^2 + c^2 n^2) over unit directions, with semi-axes (1, 1.5, 2) fixed to the operator defaults

Chart reproduced numerically against the shipped implementation: fwd max 0.28 of tol, 0.0% of oracle beyond tol (tol floor 2.5 x 0.07336).

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.