Coil

Coil is a swept surface, given by a parametrisation, immersed.

aperiodic def-parametric immersed implemented swept tradition-classical

Formula

x(u, v)

\cos\left(v\right) - 0.45 \cos\left(u\right) \cos\left(v\right) + \frac{0.18 \sin\left(u\right) \sin\left(v\right)}{\sqrt{1.16}}

y(u, v)

\sin\left(v\right) - 0.45 \cos\left(u\right) \sin\left(v\right) - \frac{0.18 \sin\left(u\right) \cos\left(v\right)}{\sqrt{1.16}}

z(u, v)

0.4 v + \frac{0.45 \sin\left(u\right)}{\sqrt{1.16}}

Properties

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

Definition

tube radius 0.45 about the helix (cos v, sin v, 0.4 v), one turn, the cross-section circle riding in the helix's Frenet normal plane -- the operator defaults. The builder fits its output to the 2 m cube; verification applies the same fit

Sources

Chart reproduced numerically against the shipped implementation: fwd max 0.40 of tol, 0.0% of oracle beyond tol (tol floor 2.5 x 0.03515).

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.
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".
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.
The helicoid is a classical minimal and ruled surface (J. B. C. Meusnier, 1776); the hyperbolic helicoid, conical seashell and twisted-sphere forms here are standard parametric surfaces. See A. Gray, E. Abbena, S. Salamon, "Modern Differential Geometry of Curves and Surfaces with Mathematica" (3rd ed., 2006), and J. Meier's gallery (3d-meier.de).
R. Ferreol, "Coil (Serpentin)", Encyclopedie des formes mathematiques remarquables, mathcurve.com; local mirror chapter ch1233_serpentin_2 (S:/data/math_art/references/websites/mathcurve/book/surfaces/ch1233_serpentin_2.md).