Swallowtail (quartic discriminant)

Swallowtail (quartic discriminant) is an algebraic surface, defined by an implicit equation, immersed with singularities.

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algebraic aperiodic def-implicit implemented singular tradition-classical

Formula

256 z^{3} - 128 x^{2} z^{2} + 144 x y^{2} z - 27 y^{4} + 16 x^{4} z - 4 x^{3} y^{2} = 0

Properties

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

Definition

The swallowtail of singularity theory: the discriminant surface of the general quartic polynomial in one variable (parameter space coordinates where the quartic has a repeated root), one of the elementary catastrophes.

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).

E. W. Weisstein, "Peano Surface", "Chair", "Crossed Trough", "Handkerchief Surface", "Hunt's Surface", "Kiss Surface", "Menn's Surface", "Miter Surface", "Nordstrand's Weird Surface", "Tooth Surface", MathWorld -- A Wolfram Web Resource, mathworld.wolfram.com.
T. Nordstrand, "Weird Cube" and the tooth/pillow object, per the MathWorld pages (Nordstrand 1997).
G. Peano's 1899 counterexample; cf. the MathWorld page's history of the criterion it defeated.
O. Labs, 'Advent calendar 2002, No. 9', The Algebraic Surface Homepage, algebraicsurface.net (Wayback rescue, captures to 2017-10-01; local mirror: algsurf/book/.../ch027_no_09.md).