Coordinates and related types


There are a large number of coordinate types used in crystallographic calculations, and a significant number of other small data types also in common use. These types, their uses, and the conversions between them will be discussed in the following sections.

The coordinate types are divided into two classes:

In addition, data types are defined for map gradients and curvatures and for anisotropic atomic displacement parameters (B or U values).

3-vector and matrix types.

All of the coordinate types are derived from a template class for a vector of 3 values; clipper::Vec3<T>. Generally T will be the floating point type, but in some cases vectors of int are used. Vec3<> also defines various simple arithmetic operations, including addition, subtraction, negation, and scaling. Any coordinate type may be constructed from a Vec3<> by using an explicit constructor. (Implicit conversion is not allowed, since this would allow accidental type conversion between coordinate types). The elements of the vector may be accessed as v[0], v[1], and v[2], however the coordinate types all define type-specific names for the members.

Similarly, clipper::Mat33<T> provides a template class for a 3x3 matrix of values. Matrix-vector and matrix-matrix products are defined. If a vector pre-multiplies a matrix, it is assumed that the vector is a row rather than column vector.

Symmetric matrices are common on crystallography, so a type is defined for these; clipper::Mat33sym<T>. This does not define any operations except conversion to or from Mat33<T>. This form is therefore used mainly for packed storage of some quantities, such as anisotropic U values.

Rotation-translation operators are provided by the clipper::RTop<T> template class. A rotation-translation operator consists of a rotation defined by a Mat33<T>, and a translation defined by a Vec3<T>. The matrix and vector components may be accessed through the rot() and trn() methods. RTop's may be applied to vectors or combined using the '*' operator.

Coordinate types

Reciprocal space coordinate types

The integral index (sometimes referred to as a Miller index) of a reflection. It has components h, k, and l.
Reciprocal space fractional coordinate. This is identical to an HKL, except that the values are floating point. This would commonly be used in rotation function calculations when interpolation is required in reciprocal space.
Reciprocal space orthogonal coordinate. This is a coordinate defined in inverse Angstroms along three orthogonal axes. The exact convention for selection of these axes is defined by the clipper::Cell class.

Real space coordinate types

Orthogonal coordinate in Angstroms. This is a coordinate defined in Angstroms along three orthogonal axes. The exact convention for selection of these axes is defined by the clipper::Cell class.
Fractional (crystal) coordinate. This is defined as the fractional position in the unit cell, with each component traversing a unique cell over the range (0..1).
Map coordinate. This is a floating point coordinate representing a position in the frame of a map. In the case of a crystallographic map, this will simply be the fractional coordinate scaled by the grid sampling of the unit cell (not the asymmetric unit). However in most cases Coord_frac is used in preference for crystallographic maps. In the case of a non-crystallographic map, fractional coordinates are undefined, but the NXmap will define a conversion between orthogonal and map coordinates.
Map grid coordinate. This is the integer version of a Coord_map, and represents the index of a grid point in a map. The origin is always at the origin of the coordinate frame of the map, thus for crystallographic maps the origin is the crystallographic origin. For non-crystallographic maps, the origin is the first grid point in the map.

Derivative types

Derivatives and curvatures may be with respect to orthogonal, fractional, or map coordinates. When performing calculations on a crystallographic map, derivatives are usually with respect to fractional coordinates, however map coordinates may be used if preferred. The resulting derivatives may then be converted to orthogonal coordinates if necessary. When performing calculations on a non-crystallographic map, derivatives are calculated with respect to map coordinates.

Derivatives are stored as Vec3<> with the template type matching the type of the map. Curvatures are stored as a Mat33<>.

Operator Types

Two types of rotation-translation operators are defined as special cases of RTop<> with a floating point template type.

This is a rotation-translation operator defined on fractional coordinates. A Crystallographic symmetry operator is an example of this type, and in fact Symop is derived from this class.
This is a rotation-translation operator defined on orthogonal coordinates. A non-crystallographic symmetry operator is an example of this type.

Other types

Anisotropic atomic displacement parameters (ADPs) are also defined. These are derived from Mat33sym<> with a floating point template type. There are two types:

Anisotropic atomic displacement parameters defined on orthogonal Angstrom coordinates, as used in PDB files. The terms have units of inverse angstroms squared.
Anisotropic atomic displacement parameters defined on fractional coordinates.

Conversion of coordinates and related types

Coordinates and related types are converted by the following means: