Daniel Phillips
by Daniel Phillips

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Rotation matrices can be built by combining basic rotations in X, Y and Z (see Wikipedia), but it’s also possible to describe them by setting values in the matrix directly. I’ve recently found this useful as a quick way to create a matrix to convert from a space where the X and Y axes are flipped.

A rotation matrix can be built using three orthogonal vectors which define the original orientation of the X, Y and Z axes in the new coordinate space.

\(R = \begin{bmatrix} A_x & B_x & C_x & 0 \\ A_y & B_y & C_y & 0 \\ A_z & B_z & C_z & 0 \\ 0 & 0 & 0 & 1 \end{bmatrix}\) where \(A\), \(B\) and \(C\) are unit vectors.

For the identity matrix (and hence no rotation), \(A = \begin{pmatrix} 1 & 0 & 0 \end{pmatrix}\), \(B = \begin{pmatrix} 0 & 1 & 0 \end{pmatrix}\) and \(C = \begin{pmatrix} 0 & 0 & 1 \end{pmatrix}\).

An Example

This only made sense to me after trying a few examples.

Rotating 90 degrees around the X axis results in an orientation where:

  • \(X_{orig}\) is the original orientation of the X axis. \(X_{orig}\) is equivalent to X because rotation was around the X axis. \(X_{orig} = \begin{pmatrix} 1 & 0 & 0 \end{pmatrix}\)
  • \(Y_{orig}\) is the original orientation of the Y axis. It is now pointing along the new Z axis. \(Y_{orig} = \begin{pmatrix} 0 & 0 & 1 \end{pmatrix}\)
  • \(Z_{orig}\) is the original orientation of the Z axis. It is now pointing along the negative Y axis. \(Z_{orig} = \begin{pmatrix} 0 & -1 & 0 \end{pmatrix}\)

Putting these three vectors into the matrix above gives the following rotation matrix: \(R = \begin{bmatrix} 1 & 0 & 0 & 0 \\ 0 & 0 & -1 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 \end{bmatrix}\)

To verify this, here’s the general matrix for rotation around X, where \(\theta\) is the rotation angle: \(R_x(\theta) = \begin{bmatrix} 1 & 0 & 0 \\ 0 & \cos \theta & -\sin \theta \\ 0 & \sin \theta & \cos \theta \end{bmatrix}\)

Setting \(\theta = 90° \) gives: \(R = \begin{bmatrix} 1 & 0 & 0 & 0 \\ 0 & 0 & -1 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 \end{bmatrix}\)

Which is the same matrix that we originally created.