Rigid Motions in Three Dimensions
\[\mathbb{R}^3\]
maybe rotations, reflections or translations, or any combination of these.The linear rigid motions can be classified by the fixed points of the motion. Let
\[W\]
be the subspace consisting of the set of vectors in \[\mathbb{R}^3\]
left unchanged by the motion.If
\[dim(W)=3\]
then every vector in \[\mathbb{R}^3\]
is unchanged by the motion, and the motion is the identity.If
\[dim(W)=2\]
then \[W\]
must be a plane, and the rigid motion fixes basis vectors of the plane. If vectors perpendicular to the plane are left unchanged then the motion would be the identity. Since angles are preserved, the image of this perpendicular vector must be its reflection in the plane and we can write the transformation matrix as \[Q= \left( \begin{array}{ccc} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & -1 \end{array} \right) \]
.If
\[dim(W)=1\]
then \[W\]
must be a line, and the rigid motion is a rotation about this line as axis. Rotation clockwise about the \[x\]
- axis by the angle \[\theta\]
is represented by the matrix \[R_x ( \theta )= \left( \begin{array}{ccc} 1 & 0 & 0 \\ 0 & cos \theta & - sin \theta \\ 0 & sin \theta & cos \theta \end{array} \right)\]
.Rotation clockwise about the
\[y\]
- axis by the angle \[\beta\]
is represented by the matrix \[R_y ( \beta )= \left( \begin{array}{ccc} cos \beta & 0 & -sin \beta \\ 0 & 1 & 0 \\ sin \beta & 0 & cos \beta \end{array} \right)\]
.Rotation clockwise about the
\[z\]
- axis by the angle \[\theta\]
is represented by the matrix \[R_y ( \gamma )= \left( \begin{array}{ccc} cos \gamma & -sin \gamma & 0 \\ sin \gamma & cos \gamma & 0 \\ 0 & 0 & 1 \end{array} \right)\]
.The general rotation is a combination of these rotations and the general reflection is a combination of the reflection matrix and rotation matrices.
If
\[dim(W)=0\]
then the rigid motion is a translation along some line, or a translation followed by a rotation or reflection. 
