select points within circles within square perimeter - r

Question

I have a vector of coordinates where each row designates the centre of a circle:
x <- runif(5,0,2)
y <- runif(5,0,2)
As you can see the circles centres are all found within the square (0,2).
Each circle has a radius 0.2. I want to randomly shift the centre of the circles within the bounds of the original circle. I figured I could do this:
radii <- (sample(20,5,replace=TRUE))/100
angles <- sample(360,5,replace=TRUE)
newx <- x + radii*(cos(angles))
newy <- y + radii*(sin(angles))
However, I realise that doing this I could technically get circle centres that fall outside of the square (0,2). I could try and write a loop that rejects newx and newy values that are negative. But have to do this for 10s of thousands of rows and worried about the speed of this. Is it possible to run this conditional coordinate shift without resorting to a loop?
My rule set is as follows:
pick a new circle centre for each centre.
The new centres must fall within the area of each circle (radius 0.2 distance from the original centre)
The new centres must lie within the original square.
If a centre meets the border of the circle it should be reflected as of the law of reflection (be reflected the remaining length of the random radius distance selected)

Answer 1

Something like this:
#lets do only one point first
x <- runif(1,0,2)
y <- runif(1,0,2)
randomwalk <- function (pos) {
x <- pos[1]
y <- pos[2]
radius <- (sample(20,1,replace=TRUE))/100
angle <- sample(360,1,replace=TRUE)
newx <- x + radius*(cos(angle))
newy <- y + radius*(sin(angle))
if (newy > 2) { #check the geometric calculations
r2 <- (2-y)/sin(angle)
hitx <- x + r2*(cos(angle))
hity <- 2
newx <- hitx + (radius-r2)*sin(angle)
newy <- hity - (radius-r2)*cos(angle)
}
#implement other borders yourself
#and include a check, which border is hit first
#and include the possibility for multiple hits
#(e.g., left border and then top border)
cbind(newx,newy)
}
resx <- vector(50,mode="numeric")
resy <- vector(50,mode="numeric")
res <- cbind(resx,resy)
res[1,] <- cbind(x,y)
for (i in 2:50) {
res[i,] <- randomwalk(res[i-1,])
}
I suspect this still contains some geometric errors, but don't have time to check.

Answer 2

The functions inpip and inout from package splancs is quite useful; they can be used to check if points fall inside a polygon. You just need a matrix with 2 columns which represents any polygon (such as a square). This functions are made to be fast, using C and Fortran programs.
If your square is:
square <- cbind(c(0, 10, 10, 0), c(0, 0, 10, 10)) # In case side = 10
Then create all the new centers (I suggest using runif instead of sample for the radii and angle, but that's up to you). Then check if those centers fall inside the square with one line:
inside <- inout(newCenters, square)
newCenters <- newCenters[inside]
And afterwards you should do all the necessary steps to recreate the newCenters that where selected out, as many times as needed until they fall inside the square. Note that this needs a while loop (or equivalent).
Note also that in the same package (splancs) there is this function csr that create random points inside a polygon. So in principle you could cut a piece of every circumference that falls outside the square and then use the resulting polygons (the cut circles) as input to this function. This can become slow because you have to use a loop (or a lapply maybe) for all cut circles.
As a last idea, maybe you can combine the two strategies. First use your initial idea to all circumferences that fall completely inside the square (or equivalently, all the centers that are at a distance of 2 or more from the perimeter). Then use the csr function for all the rest of the circles.
Hope this helps!