Coordinate transformations are commonly used in crystallography to characterize the crystal symmetry. The four symmetry operations that can occur in three-dimensional geometry are rotation, reflection, inversion, and translation. This Demonstration shows rotation and rotation with inversion (roto-inversion). The original coordinate system (labeled
) is colored blue and the new coordinate system (labeled
) is colored orange. The black arrow is the axis about which the rotation occurs; the direction of the arrow is controlled with the
is the angle between the black arrow and
is the angle in the
plane between the arrow and the
(both angles are in radians). Rotation operations are described as "
refers to the number of steps to complete a full rotation. For example, a fourfold rotation means 4 steps of
for a full
rotation about the axis.