# tori: Sample (with noise) from tori In tdaunif: Uniform Manifold Samplers for Topological Data Analysis

## Description

These functions generate uniform samples from configurations of tori of primary radius 1 in 3-dimensional space, optionally with noise.

## Usage

 ```1 2 3 4 5``` ```sample_torus_tube(n, ar = 2, sd = 0) sample_tori_interlocked(n, ar = 2, sd = 0) sample_torus_flat(n, ar = 1, sd = 0) ```

## Arguments

 `n` Number of observations. `ar` Aspect ratio for tube torus (ratio of major and minor radii) or flat torus (ratio of scale factors). `sd` Standard deviation of (independent multivariate) Gaussian noise.

## Details

The function `sample_torus_tube()` uses the tubular parameterization into 3-dimensional space documented at MathWorld.

The function `sample_torus_flat()` uses a flat parameterization (having zero Gaussian curvature) into 4-dimensional space, as presented on Wikipedia.

The function `sample_tori_interlocked()` samples from two tubular tori interlocked in the same way as `sample_circles_interlocked()`.

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

## References

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

## Examples

 ```1 2 3 4 5 6 7 8 9``` ```set.seed(33183L) # torus tube embedding in 3-space x <- sample_torus_tube(120, sd = .05) pairs(x, asp = 1, pch = 19, cex = .5) # torus flat embedding in 4-space x <- sample_torus_flat(120, sd = .05) pairs(x, asp = 1, pch = 19, cex = .5) ```

tdaunif documentation built on Oct. 26, 2020, 5:08 p.m.