These functions generate uniform samples from a sphere of radius 1 and dimension 2 in 3-space, or in arbitrary dimension in 1-higher-dimensional space, optionally with noise.
sample_2hemisphere(n, bins = 1L, sd = 0)
sample_2sphere(n, bins = 1L, sd = 0)
sample_sphere(n, dim = 1, sd = 0)Arguments
- n
Number of observations.
- bins
Number of intervals per dimension to stratify by. Default set to 1, which generates a uniform sample.
- sd
Standard deviation of (independent multivariate) Gaussian noise.
- dim
Dimension of the sphere.
Details
The function sample_sphere() is adapted from sphereUnif() in the TDA
package. It uses stats::rnorm() to sample from a multivariate Gaussian and
normalizes the resulting coordinates.
The function sample_2hemisphere() uses an area-preserving parameterization
of the upper hemisphere, and sample_2sphere() uses two of these samples,
one reflected over the horizontal plane, to produce a sample from a sphere.
The parameterization was derived through the method for sampling 2-manifolds
as described by Arvo (2001).
References
J Arvo (2001) Stratified Sampling of 2-Manifolds. SIGRAPH 2001 (State of the Art in Monte Carlo Ray Tracing for Realistic Image Synthesis), Course Notes, Vol. 29. https://www.cs.princeton.edu/courses/archive/fall04/cos526/papers/course29sig01.pdf
Examples
set.seed(50253L)
# 1-sphere in 2-space
x <- sample_sphere(120, dim = 1, sd = .1)
pairs(x, asp = 1, pch = 19, cex = .5)
# 2-sphere in 3-space
x <- sample_sphere(120, dim = 2, sd = .1)
pairs(x, asp = 1, pch = 19, cex = .5)
# 3-sphere in 4-space
x <- sample_sphere(120, dim = 3, sd = .1)
pairs(x, asp = 1, pch = 19, cex = .5)
# 4-sphere in 5-space
x <- sample_sphere(120, dim = 4, sd = .1)
pairs(x, asp = 1, pch = 19, cex = .5)
