Helico-Conical Surface

Helico-Conical Surface is a swept surface, given by a parametrisation, immersed.

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aperiodic def-parametric immersed implemented swept

Formula

x(u, v)

v u \cos\left(v\right)

y(u, v)

v u \sin\left(v\right)

z(u, v)

4 v \left(1 - u^{2}\right)

Properties

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

Definition

parabolic generatrix (profile power 2), one turn and height 4 fixed to the operator defaults; the curve is rotated AND scaled by v at once, so each u-line is a conical spiral

Sources

Chart reproduced numerically against the shipped implementation: fwd max 0.42 of tol, 0.0% of oracle beyond tol (tol floor 2.5 x 0.1264).

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.
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).