Spherical Helicoid
Spherical Helicoid is a swept surface, given by a parametrisation, immersed.
Open Spherical Helicoid in the interactive viewer →
aperiodic def-parametric immersed implemented swept tradition-gallery
Properties
- Family
- swept
- Given by
- a parametrisation
- Curvature
- no condition imposed
- Periodicity
- not periodic
- Ends
- 0
- Embedding
- immersed
- Fidelity
- exact
Definition
No chart is stored: the meridian is a geodesic found by two quadratures (the builder cumulatively integrates theta'(s) and z'(s) with the trapezoid rule); neither integral is elementary, so no closed-form chart exists to store. The shipped implementation is authoritative; an unverified transcription would silently define a different surface.
Sources
Derived from the Math Art generator registries and cross-referenced to the literature; not transcribed from any gallery's compilation.
- 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.
- 3DXM Virtual Math Museum, Surfaces gallery.