Rigid Body Transformations
- represents the orientation of frame with respect to frame .
- represents the translation of frame with respect to frame .
- represents the homogeneous transformations of frame with respect to frame .
Where, in 3d, is a 3 3 matrix, is a vector.
- represent rotations about the axis (right-hand rule) by angle
- Post-multiply successive transformations about intermediate frames
- Pre-multiply successive transformations about fixed frames
Composition of homogeneous transformations follows the rules of rotation matrices.
- Homogeneous transformations
Skew Symmetric Matrices
is the skew symmetrix matrix.
For any skew symmetrix matrix .
The rotation in 2d can be viewed as the complex number
Rodrigues’ formula gives us an decomposition of the rotation matrix into axis and angle.
Quatornion is a 4 dimensional representation of the rotation matrix. The basis are . The quatornion is a generalization of complex number.
- Unit Quatornion Properties
Relation to angle-axis
Operation on a vector ( is representing rotation matrix )