Bretzel2

Bretzel2 is an algebraic surface, defined by an implicit equation, immersed.

Open Bretzel2 in the interactive viewer →

algebraic aperiodic def-implicit immersed implemented tradition-gallery

Formula

\frac{\left(\left(1 - x^{2}\right) x^{2} - y^{2}\right)^{2} + \frac{z^{2}}{2}}{1 + \left(x^{2} + y^{2} + z^{2}\right)} - 0.025 = 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.