Moebius Surface

Moebius Surface is an algebraic surface, defined by an implicit equation, self-intersecting.

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algebraic aperiodic def-implicit implemented self-intersecting tradition-classical

Formula

y^{3} - 2 z y^{2} + \left(x^{2} + z^{2} - 1\right) y - 2 x z \left(x + 1\right) = 0

Properties

Family: algebraic
Given by: an implicit equation
Curvature: no condition imposed
Periodicity: not periodic
Ends: 0
Embedding: self-intersecting
Fidelity: exact
Exactness: elementary

Definition

The rotoidal algebraic model of the Mobius strip: a segment carried around a circle at half the rotation rate sweeps a cubic surface that contains the classic half-twist band. Ferreol keeps it as its own chapter, separate from the Mobius-strip chapter; the strip's other concrete models already carry records (twisted-strip, meeks-mobius-strip, sudanese-mobius-band, bjorling-twisted-band).

Sources

Closed form reproduced numerically against the shipped implementation (math_art/surfaces/algebraic.py) over 240 sample points: matches oracle x 1 over 240 points (worst 0).

R. Ferreol, "Encyclopedie des formes mathematiques remarquables", mathcurve.com -- the chapters "parapluie de Cartan", "surface de Cassini", "surface de Titeica". A converted copy of the whole encyclopedia is in research/books/ mathcurve_encyclopedie_formes_mathematiques/.
H. Cartan and H. Whitney, 1957 -- the umbrella; cf. Whitney's own umbrella x^2 = y^2 z, already in the Hauser block above.
G. D. Cassini, the ovals of 1680; the surface is the classical three-dimensional generalisation of them.
G. Titeica, "Sur une nouvelle classe de surfaces", Rend. Circ. Mat. Palermo 25 (1908) 180-187 -- the affine spheres of 1907.
A. Fresnel (1821) -- the wave surface; the chapter "Fresnel's wave surface" prints the quartic and its four singular points.
A. Gray, "Modern Differential Geometry of Curves and Surfaces with Mathematica", 2nd ed., CRC Press (1997) -- named the sine surface.
R. Ferreol, Encyclopedie des formes mathematiques remarquables (mathcurve).
R. Ferreol, "Mobius surface", Encyclopedie des formes mathematiques remarquables, mathcurve.com; local mirror chapter ch1302_mobiussurface_2 (S:/data/math_art/references/websites/mathcurve/book/surfaces/ch1302_mobiussurface_2.md).