[R] mutlidimensional in.convex.hull (wasmultidimensionalpoint.in.polygon??)

[Keith Jewell]

Hi All,

I couldn't resist doing this the "right" way!
A colleague explained the vector algebra to me (thanks Martin!) and I've 
followed the structure of the Matlab code referenced below, but all errors 
are mine!

I don't pretend to great R expertise, so this code may not be optimal (in 
time or the memory issue addressed by the Matlab code), but it is a lot 
faster than the other algorithm discussed in this thread. These timings are 
on the real example data described in my previous message in this thread.

> system.time(inhull(xs,ps))  # the "right" way
   user  system elapsed
   1.34    0.07    1.41
> system.time({phull <-  convhulln(ps) # the other algorithm
+ phull2 <- convhulln(rbind(ps,xs))
+ nrp <- nrow(ps)
+ nrx <- nrow(xs)
+ outside <- unique(phull2[phull2>nrp])-nrp
+ done <- FALSE
+ while(!done){
+     phull3 <- convhulln(rbind(ps,xs[-(outside),]))
+     also.outside <- (1:nrx)[-outside][unique(phull3[phull3>nrp])-nrp]
+     outside <- c(outside,also.outside)
+     done <- length(also.outside)==0
+ }})
   user  system elapsed
  15.09    0.09   15.22

Although I really must move on now, if anyone has comments, criticisms, 
suggestions for improvement I would be interested.

Enjoy!

Keith Jewell
----------------
inhull <- function(testpts, calpts, hull=convhulln(calpts), 
tol=mean(mean(abs(calpts)))*sqrt(.Machine$double.eps)) {
#++++++++++++++++++++
# R implementation of the Matlab code by John D'Errico 04 Mar 2006 (Updated 
30 Oct 2006)
# downloaded from 
http://www.mathworks.com/matlabcentral/fileexchange/10226-inhull
# with some modifications and simplifications
#
# Efficient test for points inside a convex hull in n dimensions
#
#% arguments: (input)
#%  testpts - nxp array to test, n data points, in p dimensions
#%       If you have many points to test, it is most efficient to
#%       call this function once with the entire set.
#%
#%  calpts - mxp array of vertices of the convex hull, as used by
#%       convhulln.
#%
#%  hull - (OPTIONAL) tessellation (or triangulation) generated by convhulln
#%       If hull is left empty or not supplied, then it will be
#%       generated.
#%
#%  tol - (OPTIONAL) tolerance on the tests for inclusion in the
#%       convex hull. You can think of tol as the distance a point
#%       may possibly lie outside the hull, and still be perceived
#%       as on the surface of the hull. Because of numerical slop
#%       nothing can ever be done exactly here. I might guess a
#%       semi-intelligent value of tol to be
#%
#%         tol = 1.e-13*mean(abs(calpts(:)))
#%
#%       In higher dimensions, the numerical issues of floating
#%       point arithmetic will probably suggest a larger value
#%       of tol.
#%
# In this R implementation default 
tol=mean(mean(abs(calpts)))*sqrt(.Machine$double.eps)
#       DEFAULT: tol = 1e-6
#
# VALUE: Matlab returns a vector of TRUE (inside/on) or FALSE (outside)
#       This R implementation returns an integer vector of length n
#       1 = inside hull
#      -1 = inside hull
#       0 = on hull (to precision indicated by tol)
#--------------------------------------------------------
   require(geometry, quietly=TRUE)  # for  convhulln
   require(MASS, quietly=TRUE)      # for Null
# ensure arguments are matrices (not data frames) and get sizes
   calpts <- as.matrix(calpts)
   testpts <- as.matrix(testpts)
   p <- dim(calpts)[2]   # columns in calpts
   cx <- dim(testpts)[1]  # rows in testpts
   nt <- dim(hull)[1]    # number of simplexes in hull
# find normal vectors to each simplex
   nrmls <- matrix(NA, nt, p)         # predefine each nrml as NA, 
degenerate
   degenflag <- matrix(TRUE, nt, 1)
   for (i in  1:nt) {
    nullsp <- t(Null(t(calpts[hull[i,-1],] - 
matrix(calpts[hull[i,1],],p-1,p, byrow=TRUE))))
    if (dim(nullsp)[1] == 1) { nrmls[i,] <- nullsp
       degenflag[i] <- FALSE}}
# Warn of degenerate faces, and remove corresponding normals
   if(length(degenflag[degenflag]) > 0) 
warning(length(degenflag[degenflag])," degenerate faces in convex hull")
   nrmls <- nrmls[!degenflag,]
   nt <- dim(nrmls)[1]
# find center point in hull, and any (1st) point in the plane of each 
simplex
   center = apply(calpts, 2, mean)
   a <- calpts[hull[!degenflag,1],]
# scale normal vectors to unit length and ensure pointing inwards
   nrmls <- nrmls/matrix(apply(nrmls, 1, function(x) sqrt(sum(x^2))), nt, p)
   dp <- sign(apply((matrix(center, nt, p, byrow=TRUE)-a) * nrmls, 1, sum))
   nrmls <- nrmls*matrix(dp, nt, p)
# if  min across all faces of dot((x - a),nrml) is
#      +ve then x is inside hull
#      0   then x is on hull
#      -ve then x is outside hull
# Instead of dot((x - a),nrml)  use dot(x,nrml) - dot(a, nrml)
   aN <- diag(a %*% t(nrmls))
   val <- apply(testpts %*% t(nrmls) - matrix(aN, cx, nt, byrow=TRUE), 
1,min)
# code  values inside 'tol' to zero, return sign as integer
   val[abs(val) < tol] <- 0
   as.integer(sign(val))
}