collection of various interactive jiffy programs for
- PDB coordinate manipulations
- generation/analysis of transformation
T. Schirmer (firstname.lastname@example.org)
Several of the routines have been adopted from programs written by J. Deisenhofer.
The interface has been written by J.W. Pflugrath.
Some notes to rotation matrices and angles
Some notes to the mathematical description of rotations of rigid objects
- rotation describes motion of a rigid body in a stationary coordinate system
- the coordinate system is right-handed and orthogonal
- rotation angles refer to right-handed rotations, i.e. clock-wise when looking from the origin along the rotation axis
1. General description of a rotation
All coordinates x of the points (atoms) of the object are transformed by a matrix R to yield a set of transformed coordinates x':
2. Rotation by κ around a given axis
The direction of the axis can be given by its direction cosines or, more commonly, by polar angles (Ω, φ). Hereby, Ω is the angle formed between the axis and the z-axis and φ the angle formed by the projection of the axis onto the xy-plane and the x-axis. For more details see ref.1
3. Rotation by Eulerian angles
Three successive rotations around axes that re-orient during the process are considered. There are various conventions for the choice of the axes. Here, we use the z y' z" convention used in CCP4 programs (AMORE, PDBSET, ...).
The object is rotated
- around the z-axis by the angle α
- then around the NEW y-axis direction y' by β
- and finally around the NEW z-axis direction z" by γ
Although, this description of a general rotation appears at first glance rather complex, it can be derived mathematically in a simple way.
Consider first the component rotation matrices Y and Z for rotations around the (stationary) y and z-axis, respectively: