Tables for
Volume F
Crystallography of biological macromolecules
Edited by M. G. Rossmann and E. Arnold

International Tables for Crystallography (2006). Vol. F, ch. 11.4, pp. 228-229   | 1 | 2 |

Section 11.4.4. Coordinate systems

Z. Otwinowskia* and W. Minorb

aUT Southwestern Medical Center at Dallas, 5323 Harry Hines Boulevard, Dallas, TX 75390-9038, USA, and bDepartment of Molecular Physiology and Biological Physics, University of Virginia, 1300 Jefferson Park Avenue, Charlottesville, VA 22908, USA
Correspondence e-mail:

11.4.4. Coordinate systems

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There are four natural coordinate systems used to describe a diffraction experiment, defined by the order in which the data are stored, the beam and gravity, or the beam and the goniostat axes (spindle or 2θ). These coordinate systems will be called, respectively, data, beam–gravity, beam–spindle and beam–2θ. Beam–gravity

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To visualize a diffraction pattern, beam–gravity is the coordinate system clearly preferred by human physiology. The universal preference to relate to the gravity direction is revealed by the observation that people generally perceive an image in a mirror as inverted left–right rather than top–down. Hence XdisplayF uses the beam–gravity coordinate system, except when diffraction data cannot be related to gravity.4 Data

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The first (1983) DENZO implementation used the data coordinate system to describe the beam position on the detector and to define the integration box. This is still the case in order to keep backward compatibility. Beam–spindle

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Until 1998, DENZO supported only a single-axis goniostat and used a beam–spindle coordinate system to define crystal and detector orientation and polarization. Initially, the goniostat spindle axis was assumed to be horizontal, so the direction perpendicular to the beam and spindle was described by the keyword vertical, which in reality may not relate to the gravity direction for some goniostats. The keyword rotx relates to rotation around the spindle axis, roty around the vertical axis and rotz around the beam axis. The definition of the orientation matrix in the communication file between DENZO and SCALEPACK uses an unintuitive convention: the letter y in roty relates to the first element of the vector, x in rotx relates to the second and z in rotz to the third. However, the matrix always has a positive determinant, so this convention has no impact on the handedness of the coordinate system. This unfortunate choice of convention, preserved for backward compatibility reasons, appears only in the communication file and has no significance for anybody who does not inspect the matrix. Beam–2θ

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The recent addition of a general goniostat introduced a conceptual change in the DENZO coordinate system. The data-collection axis can be oriented in any direction, so in principle rotx, roty and rotz no longer need to be defined relative to the data-collection axis. However, to keep the useful correlation between refinable parameters (crystal rotz and detector rotz being close to 100% correlated), one real and two virtual goniostats are used simultaneously in DENZO. Refinable crystal parameters (crystal rotx, roty, rotz) are still defined, as in the past, by the data-collection axis and the beam. This means that the directions of rotations defined by fit crystal rotx, roty and rotz do not rotate around the data-collection axis as the program advances from one image to another. This coordinate system changes with the change in direction of the data-collection axis. Crystal orientation is defined by three constant, perpendicular axes, which, in the current version, no longer have to be aligned with the physical crystal goniostat. However, the so-called 2 theta rotation has a fixed axis, and, if it exists, it defines the DENZO coordinate system together with the beam axis. Thus the current coordinate system in DENZO should be called beam–2θ. Fortunately for the user, the conversions between different coordinate systems are handled transparently. For example, the refined change in the crystal orientation is converted from the refined goniostat to the crystal-orientation goniostat. The movements of the physical goniostat are converted into appropriate changes in the diffraction pattern. The physical goniostat appears only to describe the data collection and, optionally, to calculate the physical goniostat angles that achieve particular crystal alignments.

The DENZO coordinate system (Gewirth, 1996[link]) is used in the definition of crystal goniostats, 2θ goniostat, Weissenberg coupling and polarization.

This discussion of the coordinate systems shows that the conceptual complexity of the program description does not result in complexity of the actual use of the program. The success of data analysis does not require a full understanding of the relations between internal DENZO goniostats and the coordinate systems. The reason for this complexity was to create a simple pattern of correlations between crystal and detector parameters in DENZO refinement. This in turn allows for simple and easy-to-understand control of the refinement process and simplifies problem diagnostics. For example: the definition of refined crystal rotx as rotation around the data-collection axis makes hardware problems when driving the spindle and shutter result only in fluctuations of crystal rotx. The constant nonzero value of the refined shifts between frames of crystal roty and rotz is a sign of misalignment of the data-collection axis. Although the program compensates for this misalignment with changes in crystal orientation, this introduces a small error in the Lorentz factor. The nature of these problems is such that they do not result in a complete failure of the experiment, but they do have an impact on the quality of the result. It is up to the experimenter and the instrument manager to assess the significance of these indications.


Gewirth, D. (1996). HKL manual. 5th ed. Yale University, New Haven, USA.

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