Labs Sextic (30 cusps)

Labs Sextic (30 cusps) is an algebraic surface, defined by an implicit equation, immersed.

Open Labs Sextic (30 cusps) in the interactive viewer →

algebraic aperiodic def-implicit immersed implemented tradition-classical

Formula

z \left(x + 0 y - z\right) \left(0.30901699437494745 x + 0.9510565162951535 y - z\right) \left(-0.8090169943749473 x + 0.5877852522924732 y - z\right) \left(-0.8090169943749475 x - 0.587785252292473 y - z\right) \left(0.30901699437494723 x - 0.9510565162951536 y - z\right) - \left(0.1424979955563914 \left(x^{2} + y^{2}\right) + 0.5699919822255656 z^{2} - 1\right)^{3} = 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 2.66e-15).

O. Labs, "A Sextic with 35 Cusps", arXiv:math/0502520 (2005) -- the D5 family (1) and Theorem 3, the real member with 30 cusps and 10 nodes shipped here.
O. Labs, "Dessins d'Enfants and Hypersurfaces with Many A_j-Singularities", J. London Math. Soc. (2) 74 (2006) 607-622; arXiv:math/0505022 -- construction (4), the folding-polynomial critical counts, Theorem 7 and the 132 = 12*11 row of Table 1.
O. Labs, Algebraic Surface Homepage, algebraicsurface.net, advent calendar 2002 Nos. 1 and 17 -- the descriptions the dodecic and Hyde rows are verified against. Mirrored locally under references/websites/algsurf/.
S. V. Chmutov, "Examples of Projective Surfaces with Many Singularities", J. Algebraic Geom. 1 (1992) 191-196 -- the nodal construction the A_j series modifies.
E. W. Hyde, "On a Surface of the Sixth Order Which Is Touched by the Axes of All Screws Reciprocal to Three Given Screws", Ann. of Math. (2) 2 (1901) 179-188 -- the surface the HYDE row rebuilds from its classical definition.
R. S. Ball, "A Treatise on the Theory of Screws", Cambridge University Press (1900) -- the pitch quadric and the canonical form of a three-system used in the derivation.