Question

I want to visualize the the Algorithm of the Polar method from two perspectives (bird’s-eye and frog’s eye view).

To draw the first step of the method I need random number of the uniform distribution in a square and a circle.

I’ve already been able to plot the circle in a square, but how to plot random numbers (z1) inside of this construct?

require (plotrix)
require (grid)
z1 = runif (100)

plot (c(-1,1), c(-1,1), type="n", asp=1)
rect(-1,-1,1,1)
draw.circle (0,0,1)

And how to change the perspective?

Answer 1

You can do without ‘rejecting’ any points. The following R function needs 3*n random numbers and will generate n randomly chosen points in a circle of radius r:

randp <- function(n = 1, r = 1) {
    if (n < 1 || r < 0) return(c())
    x <- rnorm(n)
    y <- rnorm(n)
    r <- r * sqrt(runif(n)/(x^2 + y^2))
    if (n == 1) U <- c(x, y)
    else        U <- cbind(r*x, r*y)
    return(U)
}