Orientation (geometry)

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In , the orientation, angular position, attitude, or direction of an object such as a , or is part of the description of how it is placed in the it occupies. More specifically, it refers to the imaginary 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. It may be necessary to add an imaginary , called the object's location (or position, or linear position). The location 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 location does not change when it rotates. shows that in three dimensions any orientation can be reached with a single . This gives one common way of representing the orientation using an . Other widely used methods include , , or . More specialist uses include in crystallography, in geology and on maps and signs. may also be used to represent an object's orientation. Typically, the orientation is given relative to a , usually specified by a .

# Mathematical representations

## Three dimensions

In general the position and orientation in space of a 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 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 , , or can be specified with only two values, for example two . 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 . Likewise, the orientation of a can be described with two values as well, for instance by specifying the orientation of a line 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.

## Two dimensions

In the orientation of any object (line, vector, or ) 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.

# Rigid body in three dimensions

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

## Euler angles

The first attempt to represent an orientation is attributed to . He imagined three reference frames that could rotate one around the other, and realized that by starting with a fixed reference frame and performing three rotations, he could get any other reference frame in the space (using two rotations to fix the vertical axis and another to fix the other two axes). The values of these three rotations are called .

### Taitâ€“Bryan angles

These are three angles, also known as yaw, pitch and roll, Navigation angles and Cardan angles. Mathematically they constitute a set of six possibilities inside the twelve possible sets of Euler angles, the ordering being the one best used for describing the orientation of a vehicle such as an airplane. In aerospace engineering they are usually referred to as Euler angles.

## Orientation vector

Euler also realized that the composition of two rotations is equivalent to a single rotation about a different fixed axis (). Therefore, the composition of the former three angles has to be equal to only one rotation, whose axis was complicated to calculate until matrices were developed. Based on this fact he introduced a vectorial way to describe any rotation, with a vector on the rotation axis and module equal to the value of the angle. Therefore, any orientation can be represented by a rotation vector (also called Euler vector) that leads to it from the reference frame. When used to represent an orientation, the rotation vector is commonly called orientation vector, or attitude vector. A similar method, called , describes a rotation or orientation using a aligned with the rotation axis, and a separate value to indicate the angle (see figure).

## Orientation matrix

With the introduction of matrices, the Euler theorems were rewritten. The rotations were described by referred to as rotation matrices or direction cosine matrices. When used to represent an orientation, a rotation matrix is commonly called orientation matrix, or attitude matrix. The above-mentioned Euler vector is the of a rotation matrix (a rotation matrix has a unique real ). The product of two rotation matrices is the composition of rotations. Therefore, as before, the orientation can be given as the rotation from the initial frame to achieve the frame that we want to describe. The of a non- object in ''n''-dimensional space is . Orientation may be visualized by attaching a basis of to an object. The direction in which each vector points determines its orientation.

## Orientation quaternion

Another way to describe rotations is using , also called versors. They are equivalent to rotation matrices and rotation vectors. With respect to rotation vectors, they can be more easily converted to and from matrices. When used to represent orientations, rotation quaternions are typically called orientation quaternions or attitude quaternions.

# Plane in three dimensions

## Miller indices

The attitude of a is the orientation of the line normal to the plane, and is described by the plane's . In three-space a family of planes (a series of parallel planes) can be denoted by its (''hkl''), so the family of planes has an attitude common to all its constituent planes.

## Strike and dip

Many features observed in geology are planes or lines, and their orientation is commonly referred to as their ''attitude''. These attitudes are specified with two angles. For a line, these angles are called the ''trend'' and the ''plunge''. The trend is the compass direction of the line, and the plunge is the downward angle it makes with a horizontal plane. For a plane, the two angles are called its ''strike (angle)'' and its ''dip (angle)''. A ''strike line'' is the intersection of a horizontal plane with the observed planar feature (and therefore a horizontal line), and the strike angle is the ''bearing'' of this line (that is, relative to or from ). The dip is the angle between a horizontal plane and the observed planar feature as observed in a third vertical plane perpendicular to the strike line.

# Usage examples

## Rigid body

The attitude of a rigid body is its orientation as described, for example, by the orientation of a frame fixed in the body relative to a fixed reference frame. The attitude is described by ''attitude coordinates'', and consists of at least three coordinates. One scheme for orienting a rigid body is based upon body-axes rotation; successive rotations three times about the axes of the body's fixed reference frame, thereby establishing the body's . Another is based upon , although these terms also refer to