runif_ellipsoid {uniformly}

Uniform sampling on/in ellipsoid

Description

Uniform sampling on an ellipsoid or in an ellipsoid. The sampling in an ellipsoid is available in arbitrary dimension. The sampling on an ellipsoid is available only in dimension 2 or 3.

Usage

runif_on_ellipse(n, A, r)

runif_on_ellipsoid(n, A, r)

runif_in_ellipsoid(n, A, r)

Arguments

n

number of simulations

A

symmetric positive-definite matrix defining the ellipsoid (see Details), of size 2 for runif_on_ellipse and size 2 or 3 for runif_on_ellipsoid (for size 2 these are the same functions)

r

"radius" (see Details)

Details

The ellipsoid is the set of vectors x satisfying t(x) %*% A %*% x == r^2. For example, for an axis-aligned ellipse with horizontal radius a and vertical radius b, take A=1/diag(c(a^2,b^2)) and r=1.

Value

The simulations in a matrix with n rows.

Examples

library(uniformly)
set.seed(666L)
# ellipse parameters
A <- rbind(c(2, 1), c(1, 1))
r <- 2
# plot the ellipse
x1 <- seq(-2.5, 2.5, length.out = 100)
x2 <- seq(-3, 3, length.out = 100)
z <- outer(
  x1, x2, FUN = Vectorize(function(x1, x2) t(c(x1, x2)) %*% A %*% c(x1, x2))
)
contour(x1, x2, z, nlevels = 1, levels = r^2, asp = 1, drawlabels = FALSE)
# simulations on the perimeter
sims <- runif_on_ellipse(60, A, r)
points(sims, pch = 19, col = "blue")
# simulations in the area
sims <- runif_in_ellipsoid(100, A, r)
points(sims, pch = 19, col = "green")
# 3D example ####
A <- matrix(c(5,1,1, 1,3,1, 1,1,1), ncol = 3L)
r <- 2
# draw the ellipsoid
library(misc3d)
x <- seq(-1, 1, length.out = 50)
y <- seq(-1.5, 1.5, length.out = 50)
z <- seq(-2.7, 2.7, length.out = 50)
g <- as.matrix(expand.grid(x = x, y = y, z = z))
voxel <- 
  array(apply(g, 1L, function(v) t(v) %*% A %*% v), dim = c(50, 50, 50))
isosurface <- computeContour3d(voxel, max(voxel), r^2, x = x, y = y, z = z)
drawScene.rgl(makeTriangles(isosurface, alpha = 0.3))
# simulate and plot points on ellipsoid
library(rgl)
sims <- runif_on_ellipsoid(300, A, r)
points3d(sims)