Transformations: Change of Basis Matrix

When dealing with tensors and having to transform them from one basis to another the formula to do so isn’t all that intuitive, especially when 3D programmers are used to transformation concatenations.

I’m sure any reader familiar with 3D would understand that two transformation matrices can be combined to make a new one with the same effect as the first two. This can look like so:

\begin{equation}
A * B = C
\label{eq1}
\end{equation}

In the above equation the transformations present (A B and C) would be used to transform a single vector. It should noted that this article refers to vectors as points (or more clearly vectors translating the origin). In column major format this would mean that the a given transformation would transform the column of a single vector. Given a matrix A and a vector V, to transform V into V’ the following equation would be used:

\begin{equation}
A * V = V’
\label{eq2}
\end{equation}

Similarly, if V needs to be transformed into the frame of A (which is different than being transformed by the frame of A), the following equation can be used:

\begin{equation}
A^{-1} * V = V’
\label{eq3}
\end{equation}

Say a programmer needs to take a transformation matrix and see what this would look like within another coordinate frame. An example of this would be the need to rotate a rotation matrix, or perhaps transform an intertia tensor from local to world frame. In order to apply a change of basis upon a matrix the following equation must be used, where B is called the change of basis matrix:

\begin{equation}
A’ = B * A * B^{-1}
\label{eq4}
\end{equation}

A Euclidean vector can be thought of as a linear combination of the Euclidean basis. Given a vector v:

\begin{equation}
V = \begin{bmatrix} x \\ y \\ z \end{bmatrix}
\label{eq5}
\end{equation}

And Euclidean basis as a formation of three principal vector axes scaled by scaling factors i, j and k:

\begin{equation}
E^{3} = i\begin{bmatrix} 1 \\ 0 \\ 0 \end{bmatrix}, j\begin{bmatrix} 0 \\ 1 \\ 0 \end{bmatrix}, k\begin{bmatrix} 0 \\ 0 \\ 1 \end{bmatrix}
\label{eq6}
\end{equation}

The vector V can be written as a linear combination of E^3 by replacing the i, j and k components with V’s x, y and z components:

\begin{equation}
V = x\begin{bmatrix} 1 \\ 0 \\ 0 \end{bmatrix}, y\begin{bmatrix} 0 \\ 1 \\ 0 \end{bmatrix}, z\begin{bmatrix} 0 \\ 0 \\ 1 \end{bmatrix}
\label{eq7}
\end{equation}

Following this line of thinking, a 3×3 matrix for some arbitrary rotational transformation actually contains an orthonormal basis of three principal axes. The first column represents the x axis, the second column represents the y axis, and the third column represents the z axis of a given reference frame. Any vector in E^3 can be described by a different combination of i, j and k values in any new basis. Wikipedia actually has a beautiful picture to visualize a vector within two different bases simultaneously:

Here is a simple example of showing V as a combination of the columns of some arbitrary basis matrix:

\begin{equation}
V = x\begin{bmatrix} a \\ b \\ c \end{bmatrix}, y\begin{bmatrix} d \\ e \\ f \end{bmatrix}, z\begin{bmatrix} g \\ h \\ i \end{bmatrix}
\label{eq8}
\end{equation}

Now back to equation \eqref{eq4}, in order to rotate a rotation it might be good to clearly define what this means. This means: given a rotation A it is desired to apply this rotation within the reference frame of B.

Say we have a rotation A that rotates about the x axis by θ. We would like to tilt A slightly, which would tilt the x axis slightly, and apply a rotation of θ about the new slightly tilted axis. The tilt rotation itself is given by B.

Just multiplying A by B, or B by A does not solve this problem. Multiplying A by B would translate to “rotate around the x axis, and then tilt a little bit”. Multiplying B by A would translate to: “Tilt a little bit, then spin around the x axis”. Neither of these are properly translating to “Rotating around the tilted x axis”. Here’s the translation of equation \eqref{eq4}: “Tilt the tilted axis so that it lays upon the x axis, then rotate about the x axis, and tilt once again”. This translation can also be stated in a slightly more formal manner: “Tilt A by inverse B (which aligns A’s x axis into the frame of B), rotate around B’s x axis, and tilt A back into it’s original frame”.

All that equation \eqref{eq4} is doing is transforming basis vectors of A into B’s frame, applying A, and transforming A back into A’s original reference frame. This matches the original clarified problem of “Given a rotation A it is desired to apply this rotation within the reference frame of B”. In more loose terms, this also means to rotate the rotation A by B.

Reference: GDC 2014 Slides – Math for Game Programmers – Understanding 3D Rotations by Stan Melax

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2 thoughts on “Transformations: Change of Basis Matrix

  1. Pingback: Inertia Tensor – Capsule | Randy Gaul's Game Programming Blog

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