Euclidean Transformation

The Euclidean transform consists of rotation and translation.

Rotation

Let’s think about rotation:

• We have a unit length orthogonal base $({\mathbf{e}}_1,{\mathbf{e}}_2,{\mathbf{e}}_3)$
• After a rotation it becomes $({\mathbf{e}}_1^{\prime },{\mathbf{e}}_2^{\prime },{\mathbf{e}}_3^{\prime })$
• A vector $\mathbf{a}$ has coordinates $(a_1,a_2,a_3)$ and $(a_1^{\prime },a_2^{\prime },a_3^{\prime })$ in the two systems.

Because the vector itself has not changed, we have:

$$[{\mathbf{e}}_1,{\mathbf{e}}_2,{\mathbf{e}}_3]\left[\begin{array}{c}{a}_1\\ a_2\\ a_3\end{array}\right]=[{\mathbf{e}}_1^{\prime },{\mathbf{e}}_2^{\prime },{\mathbf{e}}_3^{\prime }]\left[\begin{array}{c}{a}_1^{\prime }\\ a_2^{\prime }\\ a_3^{\prime }\end{array}\right]$$

To describe the relationship between the coordinates, both sides of the equation are multiplied by

$$\left[\begin{array}{c}{\mathbf{e}}_1^T\\ {\mathbf{e}}_2^T\\ {\mathbf{e}}_3^T\end{array}\right]$$

The matrix on the left becomes an identity matrix, so:

$$\begin{array}{rl}\left[\begin{array}{c}{a}_1\\ a_2\\ a_3\end{array}\right]=\underset{\text{rotation matrix}}{\underbrace{\left[\begin{array}{ccc}{\mathbf{e}}_1^T{\mathbf{e}}_1^{\prime }& {\mathbf{e}}_1^T{\mathbf{e}}_2^{\prime }& {\mathbf{e}}_1^T{\mathbf{e}}_3^{\prime }\\ {\mathbf{e}}_2^T{\mathbf{e}}_1^{\prime }& {\mathbf{e}}_2^T{\mathbf{e}}_2^{\prime }& {\mathbf{e}}_2^T{\mathbf{e}}_3^{\prime }\\ {\mathbf{e}}_3^T{\mathbf{e}}_1^{\prime }& {\mathbf{e}}_3^T{\mathbf{e}}_2^{\prime }& {\mathbf{e}}_3^T{\mathbf{e}}_3^{\prime }\end{array}\right]}}\left[\begin{array}{c}{a}_1^{\prime }\\ a_2^{\prime }\\ a_3^{\prime }\end{array}\right]& \triangleq \mathbf{R}{\mathbf{a}}^{\prime }\end{array}$$

The Rotation Matrix $\mathbf{R}$ consists of an inner product between the two sets of bases, describing the same vector’s coordinate relationship before and after the rotation.

Translation

For a vector $\mathbf{a}$, after rotation by $\mathbf{R}$ and a translation of $\mathbf{t}$, we get:

$${\mathbf{a}}^{\prime }=\mathbf{Ra}+\mathbf{t}$$

where $\mathbf{t}$ is the translation vector. Alternatively, if we call the original coordinate system $1$ and $2$, then we have:

$${\mathbf{a}}_1={\mathbf{R}}_{12}{\mathbf{a}}_2+{\mathbf{t}}_{12}$$

where ${\mathbf{R}}_{12}$ means “rotation of vector from system 2 to system 1”.

• Note that this is read from right to left!

The translation vector $\mathbf{t}$ corresponds to from system 1’s origin pointing to system 2’s origin. Note that:

$${\mathbf{t}}_{12}\ne -{\mathbf{t}}_{21}$$

as ${\mathbf{t}}_{12}$ needs to be taken in system 2.