Math Art
Pretzel

Pretzel

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

Open Pretzel in the interactive viewer →

algebraic aperiodic def-implicit immersed implemented tradition-gallery

Formula

((x−1)2+y2−1)⁢((x+1)2+y2−1)1+2⁢(x2+y2)⁢((x−1)2+y2−1)⁢((x+1)2+y2−1)1+2⁢(x2+y2)+2⁢z2−0.05=0\frac{\frac{\left(\left(x - 1\right)^{2} + y^{2} - 1\right) \left(\left(x + 1\right)^{2} + y^{2} - 1\right)}{1 + 2 \left(x^{2} + y^{2}\right)} \left(\left(x - 1\right)^{2} + y^{2} - 1\right) \left(\left(x + 1\right)^{2} + y^{2} - 1\right)}{1 + 2 \left(x^{2} + y^{2}\right)} + 2 z^{2} - 0.05 = 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 8.88e-16).