## Main Content

## Modtrafo

collection of various interactive jiffy programs for

- PDB coordinate manipulations
- generation/analysis of transformation

## Author

T. Schirmer (tilman.schirmer@unibas.ch)

Several of the routines have been adopted from programs written by J. Deisenhofer.

The interface has been written by J.W. Pflugrath.

## Examples

## Downloads

Modtrafo downloads

Mac OSX 10.9 Modtrafo.dmg

after installation, we suggest creating an alias, for example:

alias modtrafo='/Applications/Modtrafo.app/Contents/MacOS/modtrafo'

Older releases:

Mac OSX 10.8 modtrafo

SGI Unix modtrafo_sgi

## Some notes to rotation matrices and angles

### Some notes to the mathematical description of rotations of rigid objects

Convention:

- 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:

This can be converted further analogously to yield

which is the same result as given in ref. 1 and, e.g., the A More documentation.

### 4. References

- Evans, P.R. (2001). Rotations and rotation matrices. Acta cryst. D 57, 1355-1359
- Urzhumtsev, S. CCP4 bulletin board
- German Wiki