More on the 'Reorient' transformation

I presented the ‘Reorient’ rotation in a previous post.

I figured out three things:

• there’s an easy precise definition of this rotation, that I didn’t give;
• this rotation has a simple quaternion representation;
• this is the rotation I used to construct a mesh of a torus passing through three points.

Here is the function returning the matrix of this rotation:

# Cross product of two 3D vectors.
crossProduct <- function(v, w){
  c(
    v[2L] * w[3L] - v[3L] * w[2L],
    v[3L] * w[1L] - v[1L] * w[3L],
    v[1L] * w[2L] - v[2L] * w[1L]
  )
}
# The 'Reorient' rotation matrix. 
Reorient_Trans <- function(Axis1, Axis2){
  vX1 <- Axis1 / sqrt(c(crossprod(Axis1)))
  vX2 <- Axis2 / sqrt(c(crossprod(Axis2)))
  Y <- crossProduct(vX1, vX2)
  vY <- Y / sqrt(c(crossprod(Y)))
  Z1 <- crossProduct(vX1, vY)
  vZ1 <- Z1 / sqrt(c(crossprod(Z1)))
  Z2 <- crossProduct(vX2, vY)
  vZ2 <- Z2 / sqrt(c(crossprod(Z2)))
  M1 <- cbind(vX1, vY, vZ1)
  M2 <- rbind(vX2, vY, vZ2)
  M1 %*% M2
}

This rotation sends Axis2 to Axis1. Actually it would be more natural to take its inverse (whose matrix is obtained by transposition) but I used it in the transform3d function of the rgl package and this function requires the inverse of the transformation to be applied (oddly).

Here is how to get its inverse from a quaternion:

library(onion)
# Get a rotation matrix sending `u` to `v`; 
#   the vectors `u` and `v` must be normalized.
uvRotation <- function(u, v) {
  re <- sqrt((1 + sum(u*v))/2)
  w <- crossProduct(u, v) / 2 / re
  q <- as.quaternion(c(re, w), single = TRUE)
  as.orthogonal(q) # quaternion to rotation matrix
}

As you can see, these two rotations indeed are inverse (transpose) of each other:

u <- c(0, 0, 1)
v <- c(1, 1, 1) / sqrt(3)
Reorient_Trans(u, v)
##            [,1]       [,2]       [,3]
## [1,]  0.7886751 -0.2113249 -0.5773503
## [2,] -0.2113249  0.7886751 -0.5773503
## [3,]  0.5773503  0.5773503  0.5773503
uvRotation(u, v)
##            [,1]       [,2]      [,3]
## [1,]  0.7886751 -0.2113249 0.5773503
## [2,] -0.2113249  0.7886751 0.5773503
## [3,] -0.5773503 -0.5773503 0.5773503

Thus, uvRotation(u, v) is a rotation sending u to v. Such a rotation is not unique. We can give a precise definition of this rotation: this is the rotation which sends the plane orthogonal to u and passing through the origin to the plane orthogonal to v and passing through the origin.

And as I said in the introduction, this is the rotation I used to construct a mesh of a torus “passing through three points”. Indeed, I firstly constructed a torus having \(\{z = 0\}\) as equatorial plane, and then I mapped it with a rotation and a translation. The rotation in question is simply the uvRotation(u, v) transformation with \(u = (0,0,1)\), orthogonal to the plane \(\{z = 0\}\), and \(v\) the normal of the plane passing through the three points. And the translation is the one sending the origin to the circumcenter of the three points.