User:DavidJCobb

I'm working on a Papyrus library for working with rotations. For now, this is my scratchpad.

Scratchpad

Rotation resources

This page generates forumlae to convert from Euler angles (in any convention) to rotation matrices.

This page and this page together offer the information needed to convert from Euler angles (in any convention) to an axis-angle representation by way of rotation matrices.

This page describes how to pull right-handed Euler ZYX from a rotation matrix. It's easy to convert the math to left-handed if you use the matrix formula generator linked earlier and if you understand what atan2 does and why.

The process of converting from axis angle to quaternion (and vice versa) is one of the few rotation-related operations that Wikipedia explains in plain English. Ten bucks says a pack of PhDs will eventually come along and rewrite the article into gibberish, so that's a link to an archived version of the article as it existed when I found it.

In another rare instance of clarity, Wikipedia describes how to convert from axis-angle back to a rotation matrix. The article doesn't state whether it's working with extrinsic rotations, or what the handedness of the system is, but it seems to line up with the math at one of the previously-linked pages.

Converting any Euler sequence to a rotation matrix

Source: https://web.archive.org/web/20110721191940/http://cgafaq.info/wiki/Euler_angles_from_matrix

Corrected (Wikipedia-compatible) versions of that site's LaTeX are as follows:

"Symmetries" equation 1

  P = \begin{bmatrix} 0&1&0\\0&0&1\\1&0&0 \end{bmatrix}

"Symmetries" equation 2

  P = \begin{bmatrix} 0&0&1\\0&1&0\\1&0&0 \end{bmatrix}

"First merger" equation 1

\begin{align} {\mathrm rot}({\mathbf{xzy}_s},\theta_x,\theta_z,\theta_y) &= {\mathrm rot}(y,\theta_y)\,{\mathrm rot}(z,\theta_z)\,{\mathrm rot}(x,\theta_x) \\ &= \begin{bmatrix} c_y c_z & s_y s_x - c_y s_z c_x & s_y c_x + c_y s_z s_x \\ s_z & c_z c_x & -c_z s_x \\ -s_y c_z & c_y s_x + s_y s_z c_x & c_y c_x - s_y s_z s_x \end{bmatrix} \end{align}

"First merger" equation 2

\begin{align} {\mathrm rot}({\mathbf{xzx}_s},\theta_x,\theta_z,\theta_{x'}) &= {\mathrm rot}(x,\theta_{x'})\,{\mathrm rot}(z,\theta_z)\,{\mathrm rot}(x,\theta_x) \\ &= \begin{bmatrix} c_z & -s_z c_x & s_z s_x \\ c_{x'} s_z & c_{x'} c_z c_x - s_{x'} s_x & -s_{x'} c_x - c_{x'} c_z s_x \\ s_{x'} s_z & s_{x'} c_z c_x + c_{x'} s_x & c_{x'} c_x - s_{x'} c_z s_x \end{bmatrix} \end{align}

"First merger" equation 3

{\mathrm rot}(z,\theta_z)\,{\mathrm rot}(x,\theta_x) = \begin{bmatrix} c_z & -s_z c_x & s_z s_x \\ s_z & c_z c_x & -c_z s_x \\ 0 & s_x & c_x \end{bmatrix}