The axis is a directed line in space, along which a translation may occur, and about which rotation may occur. As an axis, this parameter cannot describe pure translation with no rotation component. As this axis can vary over time, the term 'instantaneous helical axis' (IHA) is often used. In contrast, when dealing with motion in a single 'cardinal' plane, the terms instantaneous axis of rotation (IAR), or instantaneous center of rotation (ICR), are commonly used.
The combination of a rotation about an axis and a translation in a perpendicular direction is a rotation about a parallel axis. However, a screw operation with a nonzero translation vector along the axis cannot be reduced like that. Thus the effect of a rotation combined with any translation is a screw operation in the general sense, with as special cases a pure translation. a pure rotation, and the identity. Together these are all the direct isometries in 3D.
Screw axis symmetry is invariance under a screw operation.
If φ = 360°/n for some positive integer n, then screw axis symmetry implies translational symmetry with a translation vector which is n times that of the screw operation.
Applicable for space groups is a rotation by 360°/n about an axis, combined with a translation along the axis by a multiple of the distance of the translational symmetry, divided by n. This multiple is indicated by a subscript. So, 63 is a rotation of 60° combined with a translation of 1/2 of the lattice vector, implying that there is also 3-fold rotational symmetry about this axis. The possibilities are 21, 31, 41, 42, 61, 62, and 63, and the enantiomorphous 32, 43, 64, and 65.
In any single plane, the path formed by the locations of the moving instantaneous axis of rotation (IAR) is known as the 'centroid', and is used in the description of joint motion.
In crystallography, a screw axis is a symmetry operation describing how a combination of rotation about an axis and a translation parallel to that axis leaves a crystal unchanged.
Screw axes are noted by a number, n, where the angle of rotation is 360°/n. The degree of translation is then added as a subscript showing how far along the axis the translation is, as a portion of the parallel lattice vector. For example, 21 is a 180° (two-fold) rotation followed by a translation of 1/2 of the lattice vector. 31 is a 120° (three-fold) rotation followed by a translation of 1/3 of the lattice vector. The possible screw axes are 21, 31, 41, 42, 61, 62, and 63, and the enantiomorphous 32, 43, 64, and 65.