Bretzel5

Bretzel5 is an algebraic surface, defined by an implicit equation, immersed.

Open Bretzel5 in the interactive viewer →

algebraic aperiodic def-implicit immersed implemented tradition-gallery

Formula

\left(\left(x^{2} + \frac{y^{2}}{4} - 1\right) \left(\frac{x^{2}}{4} + y^{2} - 1\right)\right)^{2} + \frac{z^{2}}{2} - 0.075 = 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).

3DXM Consortium, "Functions with compact levels in 3D-XplorMath", virtualmathmuseum.org/docs/ImplicitCompact.pdf -- the formulas and the default parameters used here. A converted copy is in research/papers/algebraic-surfaces/vmm-implicit-surfaces-compact/.
H. Karcher and R. Palais, 3D-XplorMath / Virtual Math Museum surface gallery, virtualmathmuseum.org/Surface/.
3DXM Virtual Math Museum, implicit-surfaces gallery.