Orientation in geometry
– Notion of pointing in a direction
– Changing orientation of a rigid body is the same as rotating the axes of a reference frame attached to it.
– Orientation refers to the imaginary rotation needed to move an object from a reference placement to its current placement.
– A rotation may require an imaginary translation to change the object’s position.
– Position and orientation together fully describe how an object is placed in space.
– Different methods can be used to represent orientation, such as axis-angle representation, rotation quaternions, Euler angles, and rotation matrices.
Mathematical representations in different dimensions
– The position and orientation of a rigid body can be defined relative to a local reference frame.
– At least three independent values are needed to describe the orientation of the local frame.
– The position of a point on the object can be described using three other values.
– Not all orientations are distinguishable for a rigid body with rotational symmetry.
– The orientation of a plane can be described using two values, such as the strike and dip angles.
– In two dimensions, the orientation of any object is given by a single value: the angle of rotation.
– When there are d dimensions, the specification of an orientation without rotational symmetry requires (d-1)/2 independent values.
Rotation formalisms in three dimensions
– Euler angles are one way to describe an orientation, representing three rotations around different axes.
– Tait-Bryan angles, also known as yaw, pitch, and roll, are another way to describe orientation.
– Orientation vectors, also called Euler vectors, represent rotations using a vector on the rotation axis and the angle of rotation.
– Orientation matrices, also known as rotation matrices or direction cosine matrices, describe rotations using orthogonal matrices.
– Quaternions, known as versors, are another method to describe rotations and are equivalent to rotation matrices and vectors.
Plane orientation in three dimensions
– Miller indices are used to describe the orientation of a lattice plane in three-space.
– A family of planes can be denoted by its Miller indices, which are represented by hkl.
– Different cubic crystals can have planes with different Miller indices.
– The attitude of a lattice plane is determined by the orientation of the line normal to the plane.
– The attitude of a family of planes is common to all its constituent planes.
Examples and related concepts
– The attitude of a rigid body is determined by three angles.
– Attitude coordinates are used to describe the orientation of a rigid body.
– One scheme for orienting a rigid body is based on Euler angles.
– Another scheme is based on roll, pitch, and yaw angles.
– These terms can also refer to incremental deviations from the nominal attitude.
– Angular displacement is related to the change in orientation of an object.
– Attitude control involves maintaining and adjusting the orientation of a spacecraft or other vehicles.
– Body relative direction refers to the orientation of one object relative to another.
– Directional statistics deals with the analysis of data with directional components.
– Oriented area is a concept used in differential geometry to measure the orientation of a surface. Source: https://en.wikipedia.org/wiki/Direction_(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. Two directions are obtuse if they form an obtuse angle (greater than a right angle) or, equivalently, if their scalar product or scalar projection is negative.