Hyperbolic Helicoid

Hyperbolic Helicoid is a swept surface, given by a parametrisation, immersed.

aperiodic def-parametric immersed implemented swept

Formula

x(u, v)

\frac{\sinh\left(v\right) \cos\left(2.5 u\right)}{1 + \cosh\left(u\right) \cosh\left(v\right)}

y(u, v)

\frac{\sinh\left(v\right) \sin\left(2.5 u\right)}{1 + \cosh\left(u\right) \cosh\left(v\right)}

z(u, v)

\frac{\cosh\left(v\right) \sinh\left(u\right)}{1 + \cosh\left(u\right) \cosh\left(v\right)}

torsion tau = 2.5 and extent 4 fixed to the operator defaults

Chart reproduced numerically against the shipped implementation: fwd max 0.80 of tol, 0.0% of oracle beyond tol (tol floor 2.5 x 0.004404).

Spherical helicoid: Matthias Weber, "The Spherical Helicoids", 3D-XplorMath / Virtual Math Museum documentation, https://virtualmathmuseum.org/docs/spherical_helicoid.pdf -- the helicoidal surfaces of constant Gaussian curvature K = +1, reduced via the Killing/Jacobi-field argument |J(s)| = a cos(s) to a first-order ODE for the meridian. The surface appears as an exercise in L. P. Eisenhart, "A Treatise on the Differential Geometry of Curves and Surfaces" (Ginn, 1909).
G. Darboux, "Lecons sur la theorie generale des surfaces", 1887-96
- the surfaces swept by a rigid curve. The classification of the three special motions followed here is from R. Ferreol, "Encyclopedie des formes mathematiques remarquables", mathcurve.com, chapter "surface de Darboux"; a converted copy is in research/books/mathcurve_encyclopedie_formes_mathematiques/.
Rotoid, helico-conical surface, egg box and sine torus: R. Ferreol, "Encyclopedie des formes mathematiques remarquables" (mathcurve.com), chapters "rotoide", "surface helicoconique", "boite a oeufs" and "tore sinusoidal".
Coil (serpentin): the tube whose bore is a circular helix -- R. Ferreol, ibid., chapter "serpentin"; h > 0 right-handed, h = 0 the torus, h < 0 left-handed.
Sine torus at k = 1/2: Maurice El-Milick (1947), who called it a one-sided cyclide; his model is in the Institut Henri Poincare collection.