Cassini Surface

Cassini Surface is an algebraic surface, defined by an implicit equation, immersed.

Open Cassini Surface in the interactive viewer →

algebraic aperiodic def-implicit immersed implemented tradition-gallery

Formula

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

Properties

Family: algebraic
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 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/.
G. D. Cassini, the ovals of 1680; the surface is the classical three-dimensional generalisation of them.
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. 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).