What is the equivalent of rotation matrix in Euler angle space for Quaternions?

Question

I have to load a mesh in Blender (which uses Z and Y as up and forward axes), modify it and export it (with setting up and forward axes to Y and -Z) and import it into another software which uses a coordinate system with Y up and -Z forward. Later on I export the mesh from this software and import it in Blender for rendering. Since I wanted to get the same rotation in the other software that I get in Blender, I changed the rotation matrix of the other software from Identity to the following:

[1.0, 0.0, 0.0;
0.0, 0.0, -1.0;
0.0, 1.0, 0.0]

Now I get the same exact rotation in that software that I get in Blender when using Euler angles. However, I have to actually use Quaternions and I realized that Quaternions do not have a rotation matrix. I wonder, what is the equivalent version of rotation matrix for Quaternions? Also, how can I change that equivalent-rotation-matrix of Quaternions so that I get the same rotation that I get in Blender?

Answer 1

It's not true to say quaternions don't have a rotation matrix. They are both ways of completely describing a rotation in 3D, so they are freely convertible. Each quaternion corresponds to exactly one rotation matrix, and each rotation matrix corresponds to one quaternion and its negative. Quaternions are just more compact and easier to interpolate.

To work out the rotation matrix for a quaternion, you need to take the effect of the quaternion, $q = (w, x, y, z) = (w, \textbf{v}) $ on some point $\textbf{p}$, that is, $(w, \textbf{v}) * (0, \textbf{p}) * (w, \textbf{-v}) $, and express this formula as a matrix operating on $\textbf{p}$.

Multiplying out using the rules of quaternions gives us $(w^2 - \textbf{v.v})\textbf{p} + 2( w\textbf{v}\times \textbf{p} + \textbf{(v.p)v}) $

This can be rearranged into 3 matrix terms - the first part is just a constant value down the diagonal, the second part with the cross product has values everywhere except the diagonal, and the third one is simply $\textbf{v}_m * \textbf{v}_n $ for rows m and columns n.

Adding these 3 matrices together gives you

$\begin{bmatrix} 1-2(y^2+z^2) &2(xy-zw) & 2(xz+yw)\\2(xy + zw)&1-2(x^2+z^2) & 2(yz-xw)\\2(xz-yw) & 2(yz + xw) & 1 - 2(x^2+y^2)\end{bmatrix}$

which is the conversion you need.

In your particular use case, though, you are just doing a simple rotate-x through 90 degrees, which simplifies things a lot. Remember a quaternion is $ ( cos(\theta/ 2) , sin(\theta/ 2)\textbf{a})$ where $\textbf{a}$ is your normalized axis of rotation. So the quaternion representing ±90 degrees rotation around x will be $( \frac{1}{\sqrt{2}},±\frac{1}{\sqrt{2}},0,0)$