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. A unit vector may also be used to represent an object's normal vector direction 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.

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). At least three independent values are needed to describe the orientation of this local frame. Three other values describe the position of a point on the object. All the points of the body change their position during a rotation except for those lying on the rotation axis. If the rigid body has rotational symmetry not all orientations are distinguishable, except by observing how the orientation evolves in time from a known starting orientation. For example, the orientation in space of a line, line segment, or vector can be specified with only two values, for example two direction cosines. Another example is the position of a point on the Earth, often described using the orientation of a line joining it with the Earth's center, measured using the two angles of longitude and latitude. Likewise, the orientation of a plane can be described with two values as well, for instance by specifying the orientation of a line normal to that plane, or by using the strike and dip angles.

Further details about the mathematical methods to represent the orientation of rigid bodies and planes in three dimensions are given in the following sections.

In two dimensions the orientation of any object (line, vector, or plane figure) is given by a single value: the angle through which it has rotated. There is only one degree of freedom and only one fixed point about which the rotation takes place.

When there are d dimensions, specification of an orientation of an object that does not have any rotational symmetry requires d(d − 1) / 2 independent values.

Several methods to describe orientations of a rigid body in three dimensions have been developed. They are summarized in the following sections.