[R] rgl: cylinder3d() with elliptical cross-section

[Duncan Murdoch]

My first reply to this went privately, by accident.  I've done a little 
editing to it, but mainly this is for the archives.

On 12-03-09 2:36 PM, Michael Friendly wrote:
> For a paper dealing with generalized ellipsoids, I want to illustrate in
> 3D an ellipsoid that is unbounded
> in one dimension, having the shape of an infinite cylinder along, say,
> z, but whose cross-section in (x,y)
> is an ellipse, say, given by the 2x2 matrix cov(x,y).
>
> I've looked at rgl:::cylinder3d, but don't see any way to make it
> accomplish this.  Does anyone have
> any ideas?

rgl has no way to display curved surfaces that are unbounded.  (It has 
lines and planes that adapt to the viewport.)  So you would need to make 
a finite cylinder, and it will be up to you to choose how big to make it.

The cylinder3d() function can do that, but it's not very good at 
cylinders that are straight. (This is a little embarrassing...) It sets 
up a local coordinate system based on the curvature, but if there are no 
curves, it fails, and you have to supply your own coordinates.

So here's how I would do what you want:

center <- cbind(0, 0, 1:10)  # cylinder centered on points (0, 0, z)
e2 <- cbind(1, 0, rep(0, 10)) # define the normal vectors
cyl <- cylinder3d(center, e2=e2)

# Now you have an octagonal cylinder.  Use the sides arg to cylinder3d
# if it doesn't end up smooth enough, but in most cases I've seen, 8
# is sufficient.

# Define a transformation to the x and y coordinates to give the
# elliptical shape; use it as the
# top left 2x2 matrix of a 3x3 matrix
xfrm <- matrix( c(2, 1, 0,
                                1, 3, 0,
                                0, 0, 1), 3,3, byrow=TRUE)
cyl <- transform3d(cyl, xfrm)
cyl <- addNormals(cyl)  # this makes it shade smoothly
shade3d(cyl, col="green")
decorate3d()  # show some axes for scale

Duncan Murdoch