user101089 - 2 months ago 23

R Question

To demonstrate the effect of linear transformations in 3D,

`x -> A x`

`A`

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`

`lines3d`

`c3d`

A particular transformation is given by the matrix

`A`

`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?

Answer

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.