[R] which points within an ellipsoid? Sorting data in 3d

[Kim Milferstedt]

Hello,

in a three dimensional coordinate system, I'd like to find all my 
experimental data points that fall within an ellipsoid around a fixed 
coordinate. The fixed point is defined by (x.coord.point, 
y.coord.point, z.coord.point). The coordinates of the ellipsoid are 
given by the three vectors x,y,z.

In a previous version of my code, I simply used a box instead of an 
ellipsoid to sort my data, which was really easy as I only had to 
compare each data point to three coordinates. I used something like this...

XYZ.within <- which( x.data  < x.coord.point + multipl &
                              x.data  > x.coord.point - multipl &
                              y.data  < y.coord.point + multipl &
                              y.data  > y.coord.point - multipl &
                        z.data  < z.coord.point + multipl &
                              z.data  > z.coord.point - multipl
                 )

But now I have many more coordinate sets to compare my data to. How 
can I find out in an efficient way which of the data points lie 
within the ellipsoid?

Thanks!

Kim

#mock-data for the display with rgl
data.1 <- xyz.coords(5,-2,17)
data.2 <- xyz.coords(15, -10,18)
data.3 <- xyz.coords(-19, 13,9)

#code I use to construct the ellipsoid
x.coord.point <- 4
y.coord.point <- -7
z.coord.point <- -3

radius <- 8
x.radius.multiplier <- 1
y.radius.multiplier <- 2
z.radius.multiplier <- 3

endpoint <- 350
interval <- 2
alpha <- rep(seq(0,endpoint, by 
=interval),rep((endpoint+interval)/interval,(endpoint+interval)/interval))
beta <- c(rep(seq(0,endpoint, by =interval),(endpoint+interval)/interval))

x <- 
x.coord.point+(x.radius.multiplier*radius)*cos(alpha*pi/180)*sin(beta*pi/180)
y <- y.coord.point+(y.radius.multiplier*radius)*sin(alpha*pi/180)
z <- 
z.coord.point+(z.radius.multiplier*radius)*cos(alpha*pi/180)*cos(beta*pi/180)

# using rlg to visualize the mock-data and the ellipsoid
plot.lim <- c(-20,20)
plot3d(x, y, z,
         xlim = plot.lim,
         ylim = plot.lim,
         zlim = plot.lim
         )
points3d(data.1, col ="green", size = 6)
points3d(data.2, col ="red", size = 6)
points3d(data.3, col ="blue", size = 6)

__________________________________________

Kim Milferstedt
University of Illinois at Urbana-Champaign
Department of Civil and Environmental Engineering
4125 Newmark Civil Engineering Laboratory
205 North Mathews Avenue MC-250
Urbana, IL 61801
USA
phone: (001) 217 333-9663
fax: (001) 217 333-6968
email: milferst at uiuc.edu
http://cee.uiuc.edu/research/morgenroth