Second Tractroid

Second Tractroid is a surface of revolution, given by a parametrisation, immersed.

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aperiodic def-parametric immersed implemented revolution tradition-classical

Formula

x(u, v)

\left(u - \tanh\left(u\right)\right) \cos\left(v\right)

y(u, v)

\left(u - \tanh\left(u\right)\right) \sin\left(v\right)

z(u, v)

\frac{1}{\cosh\left(u\right)}

Properties

Family: revolution
Given by: a parametrisation
Curvature: no condition imposed
Periodicity: not periodic
Ends: 0
Embedding: immersed
Fidelity: exact
Exactness: elementary

Definition

the tractrix (a(t - tanh t), a sech t) revolved about the axis PERPENDICULAR to its asymptote; a = 1 and profile reach 3 fixed to the operator defaults

Sources

Chart reproduced numerically against the shipped implementation: fwd max 0.35 of tol, 0.0% of oracle beyond tol (tol floor 2.5 x 0.02257).

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.
R. Ferreol, "Tractroid 2", Encyclopedie des formes mathematiques remarquables, mathcurve.com; local mirror chapter ch1352_tractroid_2 (S:/data/math_art/references/websites/mathcurve/book/surfaces/ch1352_tractroid_2.md).
Proposed by Ludovic Schwob, per the chapter.