Darboux Cyclide

Darboux Cyclide is a cyclide, defined by an implicit equation, immersed.

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aperiodic cyclide def-implicit immersed implemented tradition-architectural tradition-classical

Formula

\left(x^{2} + y^{2} + z^{2}\right)^{2} + \left(x^{2} + y^{2} + z^{2}\right) \cdot 0.9 x + \left(2 \left(1^{2} - 0.45^{2}\right) - 4 \cdot 1^{2}\right) \left(x^{2} + y^{2}\right) + 2 \left(1^{2} - 0.45^{2}\right) z^{2} + \left(1^{2} - 0.45^{2}\right)^{2} = 0

Properties

Family: cyclide
Given by: an implicit equation
Curvature: no condition imposed
Periodicity: not periodic
Ends: 0
Embedding: immersed
Fidelity: exact
Exactness: elementary

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 7.11e-15).

G. Darboux, "Sur une classe remarquable de courbes et de surfaces algebriques", Annales scientifiques de l'ENS (1872) -- the cyclides; the chapter "Cyclide" prints the general Darboux cyclide as (x^2+y^2+z^2)^2 + (x^2+y^2+z^2)(ax+by+cz) + P_2(x,y,z) = 0.
H. Pottmann, L. Shi and M. Skopenkov, "Darboux cyclides and webs from circles", Computer Aided Geometric Design 29 (2012) 77-97 -- the modern architectural-geometry account of the several families of circles such a surface carries.
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.
R. Ferreol, Encyclopedie des formes mathematiques remarquables (mathcurve).
G. Darboux, Principes de geometrie analytique (1917).