To demonstrate the effect of linear transformations in 3D, x -> A x
, I want to draw a cube and show its transformation under A
. For this, I need to color each face separately, and also show the vertex points and the lines that outline each face.
I can't figure out how to use distinct colors for the faces, and how to make this more general so I don't have to repeat all the steps for the result under the transformation.
what I tried:
library(rgl)
c3d <- cube3d(color=rainbow(6), alpha=0.5)
open3d()
shade3d(c3d)
points3d(t(c3d$vb), size=5)
for (i in 1:6)
lines3d(t(c3d$vb)[c3d$ib[,i],])
This gives the image below. But I don't understand how the faces are colored. And, I seem to have to use points3d
and lines3d
on the components of the c3d
shape, and don't have a single object I can transform.
A particular transformation is given by the matrix A
below, and here is how I add that to the scene,
A <- matrix(c( 1, 0, 1, 0, 2, 0, 1, 0, 2), 3, 3)
c3d_trans <- transform3d(c3d, A)
shade3d( c3d_trans )
points3d(t(c3d_trans$vb), size=5)
This gives:
Is there some way to simplify this and make it more generally useful?
In rgl
, when drawing primitive shapes, you apply colours to vertices, not faces. The faces are coloured by interpolating the colors at the vertices.
However, cube3d()
is not a primitive shape, it's a "mesh". It is drawn as 6 separate quadrilaterals. Each vertex is used 3 times.
It's not really documented, but the order the colours are used is that the first 4 are used for one face, then the next 4 for the next face, etc. If you want your colours to be rainbow(6)
, you need to replicate each colour 4 times:
library(rgl)
c3d <- cube3d(color=rep(rainbow(6), each = 4), alpha = 0.5)
open3d()
shade3d(c3d)
points3d(t(c3d$vb), size = 5)
for (i in 1:6)
lines3d(t(c3d$vb)[c3d$ib[,i],])
I'd recommend a higher alpha
value; I find the transparency a little confusing at alpha = 0.5
.
By the way, for the same purpose, I generally use a shape that looks more spherical as the baseline; I think it gives better intuition about the transformation. Here's code I have used:
sphere <- subdivision3d(cube3d(color=rep(rainbow(6),rep(4*4^4,6)), alpha=0.9),
depth=4)
sphere$vb[4,] <- apply(sphere$vb[1:3,], 2, function(x) sqrt(sum(x^2)))
open3d()
shade3d(sphere)
and this gives this shape:
which transforms to this:
A <- matrix(c( 1, 0, 1, 0, 2, 0, 1, 0, 2), 3, 3)
trans <- transform3d(sphere, A)
open3d()
shade3d(trans)
Of course, it all looks better if you can rotate it.