Clebsch Diagonal Cubic

Clebsch Diagonal Cubic is an algebraic surface, defined by an implicit equation, embedded, with C3v symmetry (a point group).

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algebraic aperiodic closed closed-orientable def-implicit embedded implemented tradition-classical tradition-physical-model

Formula

81 \left(x^{3} + y^{3} + z^{3}\right) - 189 \left(x^{2} y + x^{2} z + y^{2} x + y^{2} z + z^{2} x + z^{2} y\right) + 54 x y z + 126 \left(x y + x z + y z\right) - 9 \left(x^{2} + y^{2} + z^{2}\right) - 9 \left(x + y + z\right) + 1 = 0

Properties

Family: algebraic
Given by: an implicit equation
Curvature: no condition imposed
Periodicity: not periodic
Ends: 0
Embedding: embedded
Orientable: yes
Symmetry: C3v
Symmetry kind: point group
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 2.27e-13).

A. Clebsch, 'Ueber die Flachen vierter Ordnung, welche eine Doppelcurve zweiten Grades besitzen', J. reine angew. Math. 69 (1868); the diagonal surface.
A. Cayley and G. Salmon (1849) for the 27 lines on a cubic.