# Decompose rigid motion affine transform into parts

I have an affine transform from $R^3$ to $R^3.$ It is described as Rotation about Z axis, rotation about X axis, a translation, rotation about Z axis, and lastly a scaling (same in all 3 dimensions).

It is therefore a similarity. Now, I would like to represent this affine transform as the following composition instead: Translation, rotation about X, rotation about Y, rotation about Z, and lastly scale.

Thus, given a similarity, how do I find the 3 angles, the translation, and the scale? I know this representation is not unique, but any one should do.

If it helps, I have access to all parameters in the first representation.

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## 1 Answer

The given similarity is of the form $$S:\quad x\mapsto Ax+b\ ,$$ where the matrix $A$ and the vector $b$ are known to us. The scaling factor $\rho>0$ (if not given in advance) and the rotational part $T$ of $S$ can be read off from the matrix $A$: If $a_{\cdot1}$ is the first column of $A$ then $\rho=|a_{\cdot1}|$. The matrix $$T:={1\over\rho} A$$ is orthogonal and describes the rotational part of $S$. Now we should represent $T$ as a product of three matrices in the following way: $$T=\left[\matrix{c_3&-s_3&\cr s_3&c_3&\cr&&1\cr}\right]\left[\matrix{c_2&&s_2\cr &1&\cr -s_2&&c_2\cr}\right]\left[\matrix{1&&\cr &c_1&-s_1\cr&s_1&c_1\cr}\right]=:P\ .$$ Here $c_i=\cos\phi_i$, $s_i=\sin\phi_i$. If you compute the product $P$ of the three rotation matrices you will see that it is not a hopeless task to find the $c_i$ and $s_i$; e.g., one has $p_{31}=-s_2$.

I assume now that you have found three admissible angles $\phi_i$. It remains to find a translation vector $c$ such that $$Ax+b\ \equiv \rho\bigl(T(x+c)\bigr)\ .$$ As $A=\rho T\$ it is enough to take care of the point $x=0$. This leads to the equation $b=\rho\ Tc$ which can easily be solved for $c\>$: Since $T$ is an orthogonal matrix one gets $$c\ = {1\over\rho}\ T'b\ .$$

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That worked like a charm! –  Per Alexandersson Aug 28 '11 at 8:11