two - symmetric distance matrix r

how to calculate all pairwise distances in two dimensions (2)

The aptly named dist() will do this:

x <- matrix(rnorm(100), nrow=5)

         1        2        3        4
2 7.734978                           
3 7.823720 5.376545                  
4 8.665365 5.429437 5.971924         
5 7.105536 5.922752 5.134960 6.677726

See ?dist for more details

Say I have data concerning the position of animals on a 2d plane (as determined by video monitoring from a camera directly overhead). For example a matrix with 15 rows (1 for each animal) and 2 columns (x position and y position)

xpos<-runif(15) # x coordinates 
ypos<-runif(15) # y coordinates<-data.frame(xpos, ypos)
  rownames( = animal.ids

I want to calculate all the pairwise distances between animals. That is, get the distance from animal a (row 1) to the animal in row 2, row3...row15, and then repeat that step for all rows, avoiding redundant distance calculations. The desire output of a function that does this would be the mean of all the pairwise distances. I should clarify that I mean the simple linear distance, from the formula d<-sqrt(((x1-x2)^2)+((y1-y2)^2)). Any help would be greatly appreciated.

Furthermore, how could this be extended to a similar matrix with an arbitrarily large even number of columns (every two columns representing x and y positions at a given time point). The goal here would be to calculate mean pairwise distances for every two columns and output a table with each time point and its corresponding mean pairwise distance. Here is an example of the data structure with 3 time points:

ypos3<-runif(15)<-cbind(xpos1, ypos1, xpos2, ypos2, xpos3, ypos3)
    rownames( = letters[1:15]

Why do you compare d<-sqrt(((x1-x2)^2)+((y1-y2)^2))?

Do d^2<-(((x1-x2)^2)+((y1-y2)^2)). It will cost you much less.