Twisted Strip (solid)

Twisted Strip (solid) is a topological surface, given by a parametrisation, immersed.

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aperiodic def-parametric immersed implemented topological

Formula

x(u, v)

\left(1 + v \cos\left(\frac{u}{2}\right)\right) \cos\left(u\right)

y(u, v)

\left(1 + v \cos\left(\frac{u}{2}\right)\right) \sin\left(u\right)

z(u, v)

v \sin\left(\frac{u}{2}\right)

Properties

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

Definition

the ruled Mobius band (1 half-twist, radius 1, half-width 0.4 -- the ruled operator's defaults). The record's other construction is the solid strip with rectangular cross-section; this chart is the band's mid-surface. The u seam closes with v reversed (the half-twist)

Sources

Chart reproduced numerically against the shipped implementation: fwd max 0.28 of tol, 0.0% of oracle beyond tol (tol floor 2.5 x 0.06282).

Klein bottle: F. Klein (1882). The default classical bottle shape is built by the tube scheme of G. Franzoni, "The Klein bottle in its classical shape: a further step towards a good parametrization", arXiv:0909.5354 (2009): a tube of varying radius swept along a plane directrix, with the dumbbell-curve directrix of the paper's section 4 (which closes) as the default and its section-3 piriform directrix and the older polynomial immersion as alternatives. A converted copy is in research/papers/ surfaces-and-immersions/franzoni-2009-klein-bottle-classical-shape/.
Mobius band (the plain ruled one-sided strip): A. F. Mobius (1858) and J. B. Listing (1858), as the standard half-twist ruled parametrization.
Boy's surface: W. Boy, Math. Ann. 57 (1903), here via the R. Bryant - R. Kusner parametrization.
Cross-cap and Roman surface: two immersions of RP^2 due to J. Steiner (Rome, 1844).
Veronese surface: G. Veronese (1854-1917); see M. Berger, "Geometry Revealed", Springer 2010, p. 47. The two named projections used as the endpoints of the Steiner family here are from R. Ferreol, "Encyclopedie des formes mathematiques remarquables", mathcurve.com, chapter "surface de Veronese"; a converted copy is in research/books/ mathcurve_encyclopedie_formes_mathematiques/. Mobius band: A. F. Mobius (1858).
Sudanese Mobius band: H. B. Lawson, "Complete Minimal Surfaces in S^3", Ann. of Math. 92 (1970), 335-374; named for Sue Goodman and Daniel Asimov (cf. G. Francis, "A Topological Picturebook", Springer 1987).
G. W. Hart, "Curved, yet Straight: Stick Hyperboloids," Bridges 2023 Conference Proceedings, pp. 251-258.
E. Jannasch & J. Macnab, "The Compound Helical Cone as Kinematic Trace," Bridges 2023 Conference Proceedings, pp. 15-22.
F. A. Farris, "Spiral Ruled Surfaces," Bridges 2022 Conference
Proceedings, pp. 289-292; and F. A. Farris, "Creating Symmetry" (Princeton Univ. Press, 2015).
Antoni Gaudi (1852-1926), the ruled roof of the Sagrada Familia escoles; Hector Guimard (1867-1942). Both parametrizations, and the milk carton's, from R. Ferreol, "Encyclopedie des formes mathematiques remarquables" (mathcurve.com), chapters "surface de Gaudi", "surface de Guimard" and "berlingot".
Milk carton: H. M. Cundy & A. P. Rollett, "Mathematical Models" (1951), 185-188.