Very basic question on quaternions use for 3D rotations

Question

The quaternion $q=\cos(\pi/4) + \sin(\pi/4) \mathbf i + \sin (\pi/4)\mathbf j$ should rotate any 3D vectors around the bisecting line in the first quadrant of the $x,y$ plane $\pi/2 = 90^o.$ From Wolfram Alpha:

However, if I take the vector $(3,-4,7)$ and conjugate it:

$$\begin{align}\small (\cos(\pi/4) + \sin(\pi/4) \mathbf i + \sin (\pi/4)\mathbf j)(0 + 3\mathbf i -4 \mathbf j + 7 \mathbf k)(\cos(\pi/4) - \sin(\pi/4) \mathbf i - \sin (\pi/4)\mathbf j)\\= 0 + 9/2 \mathbf i - 6 \mathbf j - 21/2 \mathbf k\end{align}$$

But the vector $(9/2, -6,-21/2)$ doesn't look like a ninety degree rotation around $(\sin(\pi/4),\sin(\pi/4),0)$:

What obvious issue am I overlooking?

Answer 1

Of course it had to be a silly overlook, and it was.

For a $3$D vector $\mathbf v=(3,-4,7)$ to rotate around the axis of another vector $(v_1,v_2,v_3)$ using quaternions the vector $\mathbf v$ has to be thought of as pure quaternion (real part equal to zero), $\small 0 + 3 \mathbf i -4 \mathbf j + 7 \mathbf k$ conjugated by the quaternion defined by a unit vector $v_1 \mathbf i + v_2 \mathbf j + v_3 \mathbf k.$ If the axis-angle of the rotation is $\small (v_1,v_2,v_3,\phi)$ the axis vector must $\small v_1^2 + v_2^2 + v_3^2=1,$ and the rotation quaternion will be $\small \cos\left(\frac{\phi}2 \right)+v_1\sin\left(\frac{\phi}2\right) \mathbf i + v_1\sin\left(\frac{\phi}2 \right) \mathbf j + v_1\sin\left(\frac{\phi}2\right) \mathbf k.$ Therefore, $\small \cos(\pi/4) + \sin(\pi/4) \mathbf i + \sin (\pi/4)\mathbf j + 0 \mathbf k,$ in the OP, has to be normalized: this is achieved by simply turning the quaternion into $\small \mathbf q=\left(\cos\left(\frac{\pi}4\right)+ \sin\left(\frac{\pi}4\right)/\sqrt{2}\;\mathbf i + \sin\left(\frac{\pi}4\right)/\sqrt{2}\;\mathbf j + 0 \mathbf k )\right).$

Now the conjugation does what is intended:

Conjugating the vector $(3,-4,7),$

$$\begin{align} &\mathbf q\mathbf v \mathbf q'=\\ &=\Tiny \left(\cos\left(\frac{\pi}4\right)+ \frac{\sin\left(\frac{\pi}4\right)}{\sqrt{2}}\mathbf i + \frac{\sin\left(\frac{\pi}4\right)}{\sqrt{2}}\mathbf j + 0 \mathbf k )\right)(0+3\mathbf i-4\mathbf j +7\mathbf k)\left(\cos\left(\frac{\pi}4\right)- \frac{\sin\left(\frac{\pi}4\right)}{\sqrt{2}}\mathbf i - \frac{\sin\left(\frac{\pi}4\right)}{\sqrt{2}}\mathbf j + 0 \mathbf k )\right)\\ &=\Tiny \left(\cos\left(\frac{\pi}4\right)+ \frac{\sin\left(\frac{\pi}4\right)}{\sqrt{2}}\mathbf i + \frac{\sin\left(\frac{\pi}4\right)}{\sqrt{2}}\mathbf j + 0 \mathbf k )\right)(0+3\mathbf i-4\mathbf j +7\mathbf k)\left(\cos\left(-\frac{\pi}4\right) + \frac{\sin\left(-\frac{\pi}4\right)}{\sqrt{2}}\mathbf i + \frac{\sin\left(-\frac{\pi}4\right)}{\sqrt{2}}\mathbf j + 0 \mathbf k )\right)\\ &=\tiny \left(0 -\frac 1 2 +\frac{7}{\sqrt 2}\mathbf i -\frac 1 2 -\frac{7}{\sqrt 2}\mathbf j + -\frac{7}{\sqrt 2} \mathbf k )\right)=\small\color{red}{0 + 4.44975\mathbf i -5.44975\mathbf j -4.94975 \mathbf k}\\ &=\Tiny\color{blue}{\begin{bmatrix}1/2 & 1/2 & 1/\sqrt{2}\\ 1/2 & 1/2 & -1/\sqrt{2}\\ -1/\sqrt{2}& 1/\sqrt{2} &0 \end{bmatrix}}\begin{bmatrix}3\\-4\\7\end{bmatrix} \end{align}$$

The last step is the rotation matrix from angle and axis, which yields the same result as the conjugation with a simple matrix multiplication.

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Here is the Wolfram Alpha query

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I stitched together some R functions to do all these operations:

quatmul <- function(first.quaternion,second.quaternion){
  p <- first.quaternion; q <- second.quaternion
  stopifnot("Quaternions should be entered as 4-element vectors with real part in the first entry." = is.vector(p)&&is.vector(q))
  mat <- matrix(c(p[1],p[2],p[3],p[4],
                  -p[2],p[1],p[4],-p[3],
                  -p[3],-p[4],p[1],p[2],
                  -p[4],p[3],-p[2],p[1]),4,4,byrow=F)
  as.vector(round(mat%*%q,4))
}

rotmat <- function(quaternion){
  p <- quaternion
    stopifnot("Quaternions should be entered as 4-element vectors with real part in the first entry." = is.vector(p))
  mat <- matrix(c(2*(p[1]^2+p[2]^2)-1,2*(p[2]*p[3]+p[1]*p[4]),2*(p[2]*p[4]-p[1]*p[3]),
                  2*(p[2]*p[3]-p[1]*p[4]), 2*(p[1]^2+p[3]^2)-1,2*(p[3]*p[4]+p[1]*p[2]),
                  2*(p[2]*p[4]+p[1]*p[3]),2*(p[3]*p[4]-p[1]*p[2]),2*(p[1]^2+p[4]^2)-1),
                  3,3,byrow=F)
  round(mat,4)
}

quat3Drot <- function(quaternion1, quaternion2){
  #The second quaternion is just a 3D vector with a 0 in the first entry (real component).
  p <- quaternion1; q <- quaternion2
  stopifnot("Quaternions should be entered as 4-element vectors with real part in the first entry." = is.vector(p)&&is.vector(q))
  mat <- matrix(c(2*(p[1]^2+p[2]^2)-1,2*(p[2]*p[3]+p[1]*p[4]),2*(p[2]*p[4]-p[1]*p[3]),
                  2*(p[2]*p[3]-p[1]*p[4]), 2*(p[1]^2+p[3]^2)-1,2*(p[3]*p[4]+p[1]*p[2]),
                  2*(p[2]*p[4]+p[1]*p[3]),2*(p[3]*p[4]-p[1]*p[2]),2*(p[1]^2+p[4]^2)-1),
                  3,3,byrow=F)
  round(as.vector(mat%*%q[2:4]),4)
}

rotate.theta.around <- function(vector2Brotated, axis.of.rot, angle.degrees){
  r <- axis.of.rot
  q <- vector2Brotated
  stopifnot("Enter the axis of rotation and the 3D vector to be rotated both as 3-element vectors." = is.vector(r)&&is.vector(q))
  ax <- r/(sqrt(r[1]^2 + r[2]^2 + r[3]^2))
  radians <- angle.degrees/360 * 2 * pi
  p <- c(cos(radians/2), sin(radians/2)* ax)
  mat <- matrix(c(2*(p[1]^2+p[2]^2)-1,2*(p[2]*p[3]+p[1]*p[4]),2*(p[2]*p[4]-p[1]*p[3]),
                  2*(p[2]*p[3]-p[1]*p[4]), 2*(p[1]^2+p[3]^2)-1,2*(p[3]*p[4]+p[1]*p[2]),
                  2*(p[2]*p[4]+p[1]*p[3]),2*(p[3]*p[4]-p[1]*p[2]),2*(p[1]^2+p[4]^2)-1),
                  3,3,byrow=F)
  round(as.vector(mat%*%vector2Brotated),4)
}

So I can replicate the rotation matrix (in blue above):

rotmat(c(cos(pi/4),sin(pi/4)/sqrt(2),sin(pi/4)/sqrt(2),0))

A matrix: 3 × 3 of type dbl
0.5000  0.5000  0.7071
0.5000  0.5000  -0.7071
-0.7071 0.7071  0.0000

Or get the rotation in $3$D (as above in red) by using the rotation form of the quaternion:

quat3Drot(c(cos(pi/4),sin(pi/4)/sqrt(2),sin(pi/4)/sqrt(2),0),c(0,3,-4,7))
4.4497-5.4497-4.9497

Or by considering that $(\cos \frac \theta 2 + \sin \frac \theta 2 (v_1 \mathbf i + v_2 \mathbf j + v_3 \mathbf k))$ is the quaternion that allows us to rotate by $\theta$ (after conjugation) around the $3$D vector $\left(v_1,v_2,v_3\right),$ use the alternative function entering the degrees (converted to radians inside the function) to get the same result:

rotate.theta.around(c(3,-4,7), c(1/sqrt(2),1/sqrt(2),0), 90)
4.4497-5.4497-4.9497

Visualized here in reference to the vector $\small (\sin(\pi/2)/\sqrt 2,\sin(\pi/2)/\sqrt 2, 0)=(1/2,1/2,0)$:

We can try using a different axis, say the vector $(2,3,4),$ and going around the clock with two other vectors, for example, $(3,-4,7)$ and $(-4,-3,3)$:

Code here.

One final observation is that while the axis of rotation is normalized (within the function rotate.theta.around()), there is no need to normalize the vector in $3$D that we want to rotate. For example, applying the functions above to $(1,1,1)$ with and without normalizing, and rotating around $(0,1,0)$ around the clock will yield simply scaled vectors: