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Linear transformations and matrix inverse in 3D

Michael Friendly

2024-10-02

library(matlib)   # use the package
library(rgl)      # also using rgl

This vignette illustrates how linear transformations can be represented in 3D using the rgl package. It is explained more fully in this StackExchange discussion 3D geometric interpretations of matrix inverse. It also illustrates the determinant, \(\det(\mathbf{A})\), as the volume of the transformed cube, and the relationship between \(\mathbf{A}\) and \(\mathbf{A}^{-1}\).

The unit cube

Start with a unit cube, which represents the identity matrix \(\mathbf{I}_{3 \times 3}\) = diag(3) in 3 dimensions. In rgl this is given by cube3d() which returns a "mesh3d" object containing the vertices from -1 to 1 on each axis. I translate and scale this to make each vertex go from 0 to 1. These are represented in homogeneous coordinates to handle perspective and transformations. See help("identityMatrix") for details.

library(rgl)
library(matlib)
# cube, with each face colored differently
colors <- rep(2:7, each=4)
c3d <- cube3d()
# make it a unit cube at the origin
c3d <- scale3d(translate3d(c3d, 1, 1, 1),
               .5, .5, .5)
str(c3d)
## List of 6
##  $ vb       : num [1:4, 1:8] 0 0 0 1 1 0 0 1 0 1 ...
##  $ material : list()
##  $ normals  : NULL
##  $ texcoords: NULL
##  $ meshColor: chr "vertices"
##  $ ib       : num [1:4, 1:6] 1 3 4 2 3 7 8 4 2 4 ...
##  - attr(*, "class")= chr [1:2] "mesh3d" "shape3d"

The vertices are the 8 columns of the vb component of the object.

c3d$vb
##      [,1] [,2] [,3] [,4] [,5] [,6] [,7] [,8]
## [1,]    0    1    0    1    0    1    0    1
## [2,]    0    0    1    1    0    0    1    1
## [3,]    0    0    0    0    1    1    1    1
## [4,]    1    1    1    1    1    1    1    1

I want to show the transformation of \(\mathbf{I}\) by a matrix \(\mathbf{A}\) as the corresponding transformation of the cube.

# matrix A: 
A <- matrix(c( 1, 0, 1, 
               0, 2, 0,  
               1, 0, 2), 3, 3) |> print()
##      [,1] [,2] [,3]
## [1,]    1    0    1
## [2,]    0    2    0
## [3,]    1    0    2
det(A)
## [1] 2

# A can be produced by elementary row operations on the identity matrix
I <- diag( 3 )

AA <- I |>
  rowadd(3, 1, 1) |>   # add 1 x row 3 to row 1
  rowadd(1, 3, 1) |>   # add 1 x row 1 to row 3
  rowmult(2, 2)        # multiply row 2 by 2

all(AA==A)
## [1] TRUE

Define some useful functions

I want to be able to display 3D mesh objects, with colored points, lines and faces. This is a little tricky, because in rgl colors are applied to vertices, not faces. The faces are colored by interpolating the colors at the vertices. See the StackOverflow discussion drawing a cube with colored faces, vertex points and lines. The function draw3d() handles this for "mesh3d" objects. Another function, vlabels() allows for labeling vertices.

# draw a mesh3d object with vertex points and lines
draw3d <- function(object, col=rep(rainbow(6), each=4), 
                   alpha=0.6,  
                   vertices=TRUE, 
                   lines=TRUE, 
                   reverse = FALSE,
                   ...) {
  if(reverse) col <- rev(col)
    shade3d(object, col=col, alpha=alpha, ...)
    vert <- t(object$vb)
    indices <- object$ib
    if (vertices) points3d(vert, size=5)
    if (lines) {
    for (i in 1:ncol(indices))
        lines3d(vert[indices[,i],])
        }
}

# label vertex points
vlabels <- function(object, vertices, labels=vertices, ...) {
    text3d( t(object$vb[1:3, vertices] * 1.05), texts=labels, ...)
}

Show \(\mathbf{I}\) and \(\mathbf{A}\) all together in one figure

We can show the cube representing the identity matrix \(\mathbf{I}\) using draw3d(). Transforming this by \(\mathbf{A}\) is accomplished using transform3d().

open3d()
## wgl 
##   2
draw3d(c3d)
vlabels(c3d, c(1,2,3,5))

axes <- rbind( diag(3), -diag(3) )
rownames(axes) <- c("x", "y", "z", rep(" ", 3))
vectors3d(axes, frac.lab=1.2, headlength = 0.2, radius=1/20, lwd=3)

c3t<- transform3d(c3d, A) 
draw3d(c3t)
vlabels(c3t, c(1,2,3,5))

Same, but using separate figures, shown side by side

# NB: this scales each one separately, so can't see relative size
open3d()
## wgl 
##   3
mfrow3d(1,2, sharedMouse=TRUE)
draw3d(c3d)
vectors3d(axes, frac.lab=1.2, headlength = 0.2, radius=1/20, lwd=3)

next3d()    
draw3d(c3t)
vectors3d(axes, frac.lab=1.2, headlength = 0.2, radius=1/20, lwd=3)

\(A\) and \(A^{-1}\)

The inverse of A is found using `solve()

AI <- solve(A) |> print()
##      [,1] [,2] [,3]
## [1,]    2  0.0   -1
## [2,]    0  0.5    0
## [3,]   -1  0.0    1

The determinant of the matrix A here is 2. The determinant of \(\mathbf{A}^{-1}\) is the reciprocal, \(1/ \det(\mathbf{A})\). Geometrically, this means that the larger \(\mathbf{A}^{-1}\) is the smaller is \(\mathbf{A}^{-1}\).

det(A)
## [1] 2
det(AI)
## [1] 0.5

You can see the relation between them by graphing both together in the same plot. It is clear that \(\mathbf{A}^{-1}\) is small in the directions \(\mathbf{A}\) is large, and vice versa.

open3d()
## wgl 
##   4
draw3d(c3t)
c3Inv <- transform3d(c3d, solve(A))
draw3d(c3Inv)
vectors3d(axes, frac.lab=1.2, headlength = 0.2, radius=1/20, lwd=3)
vlabels(c3t, 8, "A", cex=1.5)
vlabels(c3t, c(2,3,5))

vlabels(c3Inv, 4, "Inv", cex=1.5)
vlabels(c3Inv, c(2,3,5))

Animate

The rgl graphic can be animated by spinning it around an axis using spin3d(). This can be played on screen using play3d() or saved to a .gif file using movie3d().

play3d(spin3d(rpm=15),  duration=4)

#movie3d(spin3d(rpm=15), duration=4, movie="inv-demo", dir=".")

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