Orientation (geometry)

In geometry, the orientation, attitude, bearing, direction, or angular position of an object – such as a line, plane or rigid body – is part of the description of how it is placed in the space it occupies. More specifically, it refers to the imaginary rotation that is needed to move the object from a reference placement to its current placement. A rotation may not be enough to reach the current placement, in which case it may be necessary to add an imaginary 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 above-mentioned imaginary 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. 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. Unit vector may also be used to represent an object's normal vector orientation or the relative direction between two points. Typically, the orientation is given relative to a frame of reference, usually specified by a Cartesian coordinate system. Two objects sharing the same direction are said to be codirectional (as in parallel lines). Two directions are said to be opposite if they are the additive inverse of one another, as in an arbitrary unit vector and its multiplication by -1. In general the position and orientation in space of a rigid body are defined as the position and orientation, relative to the main reference frame, of another reference frame, which is fixed relative to the body, and hence translates and rotates with it (the body's local reference frame, or local coordinate system).