## IT: Navigation mit Access Keys   ## 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.

# Online documentation

Documentation for Modtrafo is available in HTML or PDF format.

## Examples

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:

To derive the matrix of a composite rotation the component matrices are multiplied, as usual, from right to left in the order of their execution. Thus, the composite rotation matrix described by the three Eulerian angles is given by

The unknown matrix Y' (rotation around the axis y') can be derived from Y by

That is, we rotate the object back to its original orientation (by Z-1), rotate around the y-axis, and apply the z-rotation again. Thus:

This can be converted further analogously to yield

The general result is: The successive rotation around axes z, y', z" can be described by composite rotations around the fixed axes in the reverse order.

The same transformation (2) is given in ref.1, but the interpretation regarding the order of rotations is incorrect.

Finally,

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

### 4. References

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