Orientation (geometry)

In geometry, the orientation, attitude, bearing or angular position of an object – such as a line, plane or rigid body – is the rotation needed to move the object from a reference placement to its current placement. Euler's rotation theorem shows that in three dimensions any orientation can be reached with a single rotation around a fixed axis. This gives one common way of representing the orientation using an axis–angle representation. Other widely used methods include rotation quaternions, rotors, Euler angles, or rotation matrices. More specialist uses include Miller indices in crystallography, strike and dip in geology and grade on maps and signs. A unit vector may also be used to represent an object's normal vector direction or the relative direction between two points.

The orientation is part of the description of how the object is placed in the space it occupies. A rotation may not be enough to reach the current placement, in which case it may be necessary to add a translation to change the object's position (or linear position). The position and orientation together fully describe how the object is placed in space. The rotation and translation may be thought to occur in any order, as the orientation of an object does not change when it translates, and its position does not change when it rotates.

Typically, the orientation is given relative to a frame of reference, usually specified by a Cartesian coordinate system.