I'm writing code that requires rotation of objects around any point in 3D space. I've found the methods for rotating objects by the euler angles, but they all seem to rotate around the origin.
So to rotate around any point, must I first move the coordinate system by subtracting the coordinates of the rotation point from each point in the object, do the rotation, and then move the coordinate system again?
Or are the simpler, more direct (and more computationally efficient) ways to do this?
Best Answer
A 3D rotation around an arbitrary point $(x_0, y_0, z_0)$ is described by $$\left[ \begin{matrix} x^\prime \\ y^\prime \\ z^\prime \end{matrix} \right] = \left[ \begin{matrix} X_x & Y_x & Z_x \\ X_y & Y_y & Z_y \\ X_z & Y_z & Z_z \end{matrix} \right] \left[ \begin{matrix} x - x_0 \\ y - y_0 \\ z - z_0 \end{matrix} \right] + \left[ \begin{matrix} x_0 \\ y_0 \\ z_0 \end{matrix} \right]$$ which, as OP noted, first subtracts the center point, rotates around the origin, then adds back the center point; equivalently written as $$\vec{p}^\prime = \mathbf{R} \left( \vec{p} - \vec{p}_0 \right) + \vec{p}_0$$ We can combine the two translations, saving three subtractions per point – not much, but might help in a computer program. This is because $$\vec{p}^\prime = \mathbf{R} \vec{p} + \left( \vec{p}_0 - \mathbf{R} \vec{p}_0 \right)$$ In other words, you can use the simple form, either $$\left[ \begin{matrix} x^\prime \\ y^\prime \\ z^\prime \end{matrix} \right] = \left[ \begin{matrix} X_x & Y_x & Z_x \\ X_y & Y_y & Z_y \\ X_z & Y_z & Z_z \end{matrix} \right] \left[ \begin{matrix} x \\ y \\ z \end{matrix} \right] + \left[ \begin{matrix} T_x \\ T_y \\ T_z \end{matrix} \right]$$ or, equivalently, $$\left[ \begin{matrix} x^\prime \\ y^\prime \\ z^\prime \\ 1 \end{matrix} \right] = \left[ \begin{matrix} X_x & Y_x & Z_x & T_x \\ X_y & Y_y & Z_y & T_y \\ X_z & Y_z & Z_z & T_z \\ 0 & 0 & 0 & 1 \end{matrix} \right] \left[ \begin{matrix} x \\ y \\ z \\ 1 \end{matrix} \right]$$ where $$\left[ \begin{matrix} T_x \\ T_y \\ T_z \end{matrix} \right] = \left[ \begin{matrix} x_0 \\ y_0 \\ z_0 \end{matrix} \right] - \left[ \begin{matrix} X_x & Y_x & Z_x \\ X_y & Y_y & Z_y \\ X_z & Y_z & Z_z \end{matrix} \right] \left[ \begin{matrix} x_0 \\ y_0 \\ z_0 \end{matrix} \right]$$ to apply a rotation $\mathbf{R}$ around a centerpoint $(x_0, y_0, z_0)$.
The 4×4 matrix form is particularly useful if you use SIMD, like SSE or AVX, with four-component vectors; that's one reason why many 3D libraries use it. Another is that the same form can be used for projection.