Generating random points near a location

Question

In the following solution, why is there a sqrt(u)? Since u is just a random number in [0, 1) doesn't getting the sqrt just give you another smaller random number?

To generate points uniformly, randomly, and independently within a circle of radius r around a location (x0, y0), start by generating two independent uniform random values u and v in the interval [0, 1). (This is what almost every random number generator provides you.) Compute

d = r * sqrt(u)

theta = 2 * Pi * v

x = d * cos(theta)

y = d * sin(theta)

The desired random point is at location (x+x0, y+y0).

What you're asking is not really clear. You should provide more context to your question (for example, what do t and w represent?). Also, this is probably more a question for math.stackexchange.com. — ArMoraer, Mar 22 '16 at 15:54
I'm getting it from here:
http://gis.stackexchange.com/questions/25877/how-to-generate-random-locations-nearby-my-location/25883#25883

Everybody just seemed to accept the sqrt, as if it were obvious why it was there! — aleph2012, Mar 22 '16 at 16:02
This is clearer indeed. I don't have any proper explanation, only a simple thought: if you just define d as r*u, then half of your random points will fall inside a circle of radius r/2, which has an area 4 times smaller than a circle of radius r. This behavior is not consistent with a uniform random law. So I guess the sqrt is there to somehow correct this spatial repartition biais. — ArMoraer, Mar 22 '16 at 16:04
Aside from it not being clear as to what you are after, this is sounding a bit like homework. — Jeffrey Evans, Mar 22 '16 at 16:10
Basically I want to generate random points near a location within a certain distance. I thought this was pretty straightforward and came up with a solution. I did a search on stackexchange to verify it. I found that my solution was different only in the sqrt() piece - I didn't have that. I could not come up with an explanation of why it was necessary. @ArMoraer seems to have a reasonable idea though. — aleph2012, Mar 22 '16 at 16:14
Please [edit] your question to provide any requested clarifications. — PolyGeo, Mar 22 '16 at 21:22

phloem · Accepted Answer · 2016-03-22T17:22:56.520

Your first two equations choose a random distance and a random angle. Your random numbers will range from [0,1] uniformly. You can look at the rings below as limiting the choices to 0.1, 0.2, etc. If you don't alter the possible distribution of distances (left), there are many more chances to be quite close to the center. The probabilities are much more uniform in the situation on the right, weighting by the square root.

score -1 · Answer 2 · answered Mar 22 '16 at 16:07

It is not entirely clear as to what you are after. If your intent is to create a sample within a radius and exclude an inner radius, there is a function "sample.annulus" in the spatialEco R package that will create a sample given a defined inner and outer radius. The original intent of the function was to create distance lagged samples.

First we add required libraries, example data and subset point location

library(sp)
library(spatialEco)    
data(meuse)
  coordinates(meuse) <- ~x+y
   proj4string(meuse) <- CRS("+init=epsg:28992")
   xy <- meuse[2,]

Now, we run the sample.annulus function at two scales and plot results

rs100 <- sample.annulus(xy, r1=50, r2=100, n = 50, type = "random")
rs200 <- sample.annulus(xy, r1=100, r2=200, n = 50, type = "random")

plot(rs200, pch=20, col="red")
  points(rs100, pch=20, col="blue")
  points(xy, pch=20, cex=2, col="black")
  box()
  legend("topright", legend=c("50-100m", "100-200m", "source"), 
         pch=c(20,20,20), col=c("blue","red","black"))

Generating random points near a location

2 Answers2