[R] [OOPS] Re: Drawing sample from a circle

[(Ted Harding)]

On 18-Jun-10 18:18:41, Ron Michael wrote:
> Thank you so much Teds for pviding this function. Would you please
> explain the theory behind that?
> _
> Thanks,

[A]
Sampling uniformly on the circumference of the circle:

This one is simple. Given that you are sampling uniformly on the
cirumference, the position is determined by the angular distance
around the circle fron 0 to 2*pi, and this has to be uniformly
distributed. Hence u=2*pi*runif() is a uniformly distributed angle
from 0 to 2*pi. You then need to find the (x,y) coordinates of
such a point on a circle of radius "rad" centred at c(x0,y0),
which are x = rad*cos(u) + x0, y = rad*sin(u) + y0. Hence:

  csamp <- function(n,rad=1,centre=c(0,0)){
    x0 <- centre[1] ; y0 <- centre[2]
    u <- 2*pi*runif(n)
    cbind(x=rad*cos(u)+x0, y=rad*sin(u)+y0)
  }


[B]
Sampling uniformaly within the circle

Here, consider first sampling uniformly on the thin circular
domain between radius r and radius (r + rd). For small dr, we
are in effect in situation [A]; hence, given r, the angular
distances will again be uniformly distributed, so you again
use u = 2*pi*runif(n). However, the area of this circular
domain is pi*(r + dr)^2 - pi*r^2 = 2*pi*r*dr [ + dr^2 which
we neglect]. So the proportion of the points expected to fall
between r and r+dr is proportional to r*dr, i.e. the probability
density of r is proportional to r. Hence, for unrestricted
values of R, Prob[ a point at distance <= r from centre ] is
proportional to r^2, and (for an enclosing circle of unit
radius) must be equal to 1 for r=1. Hence

  Prob[ a point at distance <= r from centre ] = r^2

This is the cumulative distribution function of the random r,
and therefore r^2 is uniformly distributed: for any random
variable X, let F(x) = Prob[ X <= x ]. Then U = F(X) is
random, and

  Prob[ U <= u ] = Prob[ F(X) <= u ] = Prob[ X <= Finv(u) ]
                 = F( Finv(u) ) = u

where Finv is the inverse function of F [F(x) = u ; u = Finv(x)].

Hence U is uniformly distributed. But, for r, U = F(r) = r^2,
so r = sqrt(U) where U is uniformly distributed (as above).
Hence the line

  r <- sqrt(runif(n))

in the following [the rest being basically the same as in [A],
except that r is now random so we have the extra factor r in
cbind(x=rad*r*cos(u)+x0, y=rad*r*sin(u)+y0) ]

  Csamp <- function(n,rad=1,centre=c(0,0)){
    x0 <- centre[1] ; y0 <- centre[2]
    u <- 2*pi*runif(n)
    r <- sqrt(runif(n))
    cbind(x=rad*r*cos(u)+x0, y=rad*r*sin(u)+y0)

Hoping this helps,
Ted.

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Date: 18-Jun-10                                       Time: 21:01:17
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