rgl: ellipse3d with axes

Duncan Murdoch wrote:

The normals component contains the surface normals.  It is used to 
help in rendering the surface, but isn't much use for your purposes.
Unfortunately, I'm not familiar enough with the internals of princomp 
to tell you how to get the axes you want.

Duncan Murdoch

OK, then let me repharase it. The axes of the unit  sphere are like

axes <- matrix(
    c(0, 0, -1,   0, 0, 1,
       0, -1, 0,   0, 1,  0,
       -1, 0, 0,   1, 0, 0),  6, 3, byrow=TRUE)
taken in pairs.  I'd like to transform these coordinates the same was as 
in ellipse3d() and add them to the plot.

That's easy, but it doesn't give you the principal axes of the ellipse. 
  Just use

axes %*% chol(cov)

If you start with a unit sphere, this will give you points on its 
surface, but not the ones you want.  For those you need the SVD or 
eigenvectors.  This looks like it does what you want:

axes <- matrix(
     c(0, 0, 0, # added origin
        0, 0, -1,   0, 0, 1,
        0, -1, 0,   0, 1,  0,
        -1, 0, 0,   1, 0, 0),  7, 3, byrow=TRUE)
axes <- axes[c(1,2,1,3,1,4,1,5,1,6,1,7),]  # add the origin before each

cov <- cov(trees)
eigen <- eigen(cov)
shade3d(ellipse3d(cov, t=1, alpha=0.2, col='red'))
segments3d(axes %*% sqrt(diag(eigen$values)) %*% t(eigen$vectors))

Duncan Murdoch