These functions generate uniform samples from Klein bottles in 4-dimensional space, optionally with noise.

sample_klein_tube(n, ar = 2, sd = 0) sample_klein_flat(n, ar = 1, bump = 0.1, sd = 0)

n | Number of observations. |
---|---|

ar | Aspect ratio for Möbius tube Klein bottle (ratio of major and minor radii) or flat torus-based Klein bottle (ratio of scale factors). |

sd | Standard deviation of (independent multivariate) Gaussian noise. |

bump | Bump constant for the flat torus-based Klein bottle. |

The function `sample_klein_tube()`

uses the Möbius tube parameterization
documented at the Encyclopédie des Formes Mathématiques Remarquables.

The function `sample_klein_flat()`

uses a flat parameterization based on that
of the torus, as presented on
Wikipedia.

Both uniform samples are generated through a rejection sampling process as described by Diaconis, Holmes, and Shahshahani (2013).

P Diaconis, S Holmes, and M Shahshahani (2013) Sampling from a Manifold.
*Advances in Modern Statistical Theory and Applications: A Festschrift in
honor of Morris L. Eaton*, 102--125. doi: 10.1214/12-IMSCOLL1006

set.seed(834L) # Klein bottle tube embedding in 4-space x <- sample_klein_tube(120, sd = .05) pairs(x, asp = 1, pch = 19, cex = .5)# Klein bottle flat torus-based embedding in 4-space x <- sample_klein_flat(120, sd = .05) pairs(x, asp = 1, pch = 19, cex = .5)