In

^{''n''}. Orientation may be visualized by attaching a basis of

geometry
Geometry (from the grc, γεωμετρία; ' "earth", ' "measurement") is, with , one of the oldest branches of . It is concerned with properties of space that are related with distance, shape, size, and relative position of figures. A mat ...

, the orientation, angular position, attitude, or direction of an object such as a line
Line, lines, The Line, or LINE may refer to:
Arts, entertainment, and media Films
* ''Lines'' (film), a 2016 Greek film
* ''The Line'' (2017 film)
* ''The Line'' (2009 film)
* ''The Line'', a 2009 independent film by Nancy Schwartzman
Lite ...

, plane
Plane or planes may refer to:
* Airplane
An airplane or aeroplane (informally plane) is a fixed-wing aircraft
A fixed-wing aircraft is a heavier-than-air flying machine
Early flying machines include all forms of aircraft studied ...

or rigid body
In physics
Physics is the natural science that studies matter, its Elementary particle, fundamental constituents, its Motion (physics), motion and behavior through Spacetime, space and time, and the related entities of energy and force. " ...

is part of the description of how it is placed in the space
Space is the boundless three-dimensional
Three-dimensional space (also: 3-space or, rarely, tri-dimensional space) is a geometric setting in which three values (called parameter
A parameter (from the Ancient Greek language, Ancient Gre ...

it occupies.
More specifically, it refers to the imaginary rotation
A rotation is a circular movement of an object around a center (or point) of rotation. The plane (geometry), geometric plane along which the rotation occurs is called the ''rotation plane'', and the imaginary line extending from the center an ...

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 translation
Translation is the communication of the meaning
Meaning most commonly refers to:
* Meaning (linguistics), meaning which is communicated through the use of language
* Meaning (philosophy), definition, elements, and types of meaning discusse ...

, 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.
Euler's rotation theorem
In geometry
Geometry (from the grc, γεωμετρία; ''wikt:γῆ, geo-'' "earth", ''wikt:μέτρον, -metron'' "measurement") is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space t ...

shows that in three dimensions any orientation can be reached with a single rotation around a fixed axis
Rotation around a fixed axis is a special case of rotation
A rotation is a circular movement of an object around a center (or point) of rotation. The plane (geometry), geometric plane along which the rotation occurs is called the ''rotat ...

. This gives one common way of representing the orientation using an axis–angle representation
150px, The angle and axis unit vector define a rotation, concisely represented by the rotation vector .
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical st ...

. Other widely used methods include rotation quaternions, Euler angles
The Euler angles are three angles introduced by Leonhard Euler to describe the Orientation (geometry), orientation of a rigid body with respect to a fixed coordinate system.Novi Commentarii academiae scientiarum Petropolitanae 20, 1776, pp. 189 ...

, or rotation matrices. More specialist uses include Miller indices
Miller indices form a notation system in crystallography
Crystallography is the experimental science of determining the arrangement of atoms in crystalline solids (see crystal structure). The word "crystallography" is derived from the Greek w ...

in crystallography, strike and dip
Strike and dip refer to the orientation or ''attitude'' of a geologic feature. The ''strike line'' of a bed (geology), bed, fault, or other planar feature, is a line representing the intersection of that feature with a horizontal plane. On a ...

in geology and grade
Grade or grading may refer to:
Arts and entertainment
* Grade (band)
Grade is a melodic hardcore band from Canada, often credited as pioneers in blending metallic hardcore with the hon and melody of emo, and - most notably - the alternating scr ...

on maps and signs.
Unit vector
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ...

may also be used to represent an object's normal vector
In geometry
Geometry (from the grc, γεωμετρία; ' "earth", ' "measurement") is, with , one of the oldest branches of . It is concerned with properties of space that are related with distance, shape, size, and relative position ...

orientation.
Typically, the orientation is given relative to a frame of reference
In physics
Physics is the natural science that studies matter, its Elementary particle, fundamental constituents, its Motion (physics), motion and behavior through Spacetime, space and time, and the related entities of energy and force ...

, usually specified by a Cartesian coordinate system
A Cartesian coordinate system (, ) in a plane
Plane or planes may refer to:
* Airplane
An airplane or aeroplane (informally plane) is a fixed-wing aircraft
A fixed-wing aircraft is a heavier-than-air flying machine
Early fly ...

.
Mathematical representations

Three dimensions

In general the position and orientation in space of arigid body
In physics
Physics is the natural science that studies matter, its Elementary particle, fundamental constituents, its Motion (physics), motion and behavior through Spacetime, space and time, and the related entities of energy and force. " ...

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
Rotational symmetry, also known as radial symmetry in geometry
Geometry (from the grc, γεωμετρία; ' "earth", ' "measurement") is, with , one of the oldest branches of . It is concerned with properties of space that are related ...

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, lines, The Line, or LINE may refer to:
Arts, entertainment, and media Films
* ''Lines'' (film), a 2016 Greek film
* ''The Line'' (2017 film)
* ''The Line'' (2009 film)
* ''The Line'', a 2009 independent film by Nancy Schwartzman
Lite ...

, line segment
In geometry
Geometry (from the grc, γεωμετρία; ' "earth", ' "measurement") is, with , one of the oldest branches of . It is concerned with properties of space that are related with distance, shape, size, and relative position ...

, or vector
Vector may refer to:
Biology
*Vector (epidemiology)
In epidemiology
Epidemiology is the study and analysis of the distribution (who, when, and where), patterns and risk factor, determinants of health and disease conditions in defined pop ...

can be specified with only two values, for example two direction cosines
In analytic geometry
In classical mathematics, analytic geometry, also known as coordinate geometry or Cartesian geometry, is the study of geometry
Geometry (from the grc, γεωμετρία; ''wikt:γῆ, geo-'' "earth", ''wikt:μέτρο ...

. 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
A geographic coordinate system (GCS) is a coordinate system associated with position (geometry), positions on Earth (geographic position). A GCS can give positions:
*as Geodetic coordinates, spherical coordinate system using latitude, long ...

. Likewise, the orientation of a plane
Plane or planes may refer to:
* Airplane
An airplane or aeroplane (informally plane) is a fixed-wing aircraft
A fixed-wing aircraft is a heavier-than-air flying machine
Early flying machines include all forms of aircraft studied ...

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

Intwo dimensions
300px, Bi-dimensional Cartesian coordinate system
Two-dimensional space (also known as bi-dimensional space) is a geometric setting in which two values (called parameter
A parameter (from the Ancient Greek language, Ancient Greek wikt:παρ ...

the orientation of any object (line, vector, or plane figure
A shape or figure is the form of an object or its external boundary, outline, or external surface
File:Water droplet lying on a damask.jpg, Water droplet lying on a damask. Surface tension is high enough to prevent floating below the textile. ...

) 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 toLeonhard Euler
Leonhard Euler ( ; ; 15 April 170718 September 1783) was a Swiss mathematician
A mathematician is someone who uses an extensive knowledge of mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers ( and ) ...

. 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 Euler angles
The Euler angles are three angles introduced by Leonhard Euler to describe the Orientation (geometry), orientation of a rigid body with respect to a fixed coordinate system.Novi Commentarii academiae scientiarum Petropolitanae 20, 1776, pp. 189 ...

.
Tait–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 (Euler's rotation theorem
In geometry
Geometry (from the grc, γεωμετρία; ''wikt:γῆ, geo-'' "earth", ''wikt:μέτρον, -metron'' "measurement") is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space t ...

). 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 axis–angle representation
150px, The angle and axis unit vector define a rotation, concisely represented by the rotation vector .
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical st ...

, describes a rotation or orientation using a unit vector
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ...

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 byorthogonal matrices
In linear algebra, an orthogonal matrix, or orthonormal matrix, is a real square matrix whose columns and rows are orthonormal vectors.
One way to express this is
:Q^\mathrm Q = Q Q^\mathrm = I,
where is the transpose
In linear algebra, t ...

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 eigenvector
In linear algebra
Linear algebra is the branch of mathematics concerning linear equations such as:
:a_1x_1+\cdots +a_nx_n=b,
linear maps such as:
:(x_1, \ldots, x_n) \mapsto a_1x_1+\cdots +a_nx_n,
and their representations in vector spaces an ...

of a rotation matrix (a rotation matrix has a unique real eigenvalue
In linear algebra, an eigenvector () or characteristic vector of a Linear map, linear transformation is a nonzero Vector space, vector that changes at most by a Scalar (mathematics), scalar factor when that linear transformation is applied to i ...

).
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 configuration space of a non-symmetrical
Symmetry (from Greek συμμετρία ''symmetria'' "agreement in dimensions, due proportion, arrangement") in everyday language refers to a sense of harmonious and beautiful proportion and balance. In mathematics, "symmetry" has a more pre ...

object in ''n''-dimensional space is SO(''n'') × Rtangent vectors
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no gene ...

to an object. The direction in which each vector points determines its orientation.
Orientation quaternion

Another way to describe rotations is using rotation quaternions, 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 alattice planeIn crystallography
Crystallography is the experimental science of determining the arrangement of atoms in crystalline solids (see crystal structure). The word "crystallography" is derived from the Greek words ''crystallon'' "cold drop, frozen dr ...

is the orientation of the line normal to the plane,
and is described by the plane's Miller indices
Miller indices form a notation system in crystallography
Crystallography is the experimental science of determining the arrangement of atoms in crystalline solids (see crystal structure). The word "crystallography" is derived from the Greek w ...

. In three-space a family of planes (a series of parallel planes) can be denoted by its Miller indices
Miller indices form a notation system in crystallography
Crystallography is the experimental science of determining the arrangement of atoms in crystalline solids (see crystal structure). The word "crystallography" is derived from the Greek w ...

(''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 togeographic north
True north (also called geodetic north or geographic north) is the Cardinal direction, direction along Earth's surface towards the geographic North Pole or True North Pole.
geodesy, Geodetic north differs from ''magnetic'' north (the direction a ...

or from magnetic north
The north magnetic pole is a point on the surface of Earth's Northern Hemisphere
The Northern Hemisphere is the half of Earth
Earth is the third planet from the Sun and the only astronomical object known to harbour and support li ...

). 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'sEuler angles
The Euler angles are three angles introduced by Leonhard Euler to describe the Orientation (geometry), orientation of a rigid body with respect to a fixed coordinate system.Novi Commentarii academiae scientiarum Petropolitanae 20, 1776, pp. 189 ...

.
Another is based upon roll, pitch and yaw, although these terms also refer to
See also

*Angular displacement
Angular displacement of a body is the angle
In Euclidean geometry, an angle is the figure formed by two Ray (geometry), rays, called the ''sides'' of the angle, sharing a common endpoint, called the ''vertex (geometry), vertex'' of the angle. ...

* Attitude control
Attitude control is the process of controlling the orientation of an aerospace
Aerospace is a term used to collectively refer to the atmosphere and outer space. Aerospace activity is very diverse, with a multitude of commercial, industrial and ...

* Directional statisticsDirectional statistics (also circular statistics or spherical statistics) is the subdiscipline of statistics
Statistics is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of data. In applyin ...

* Personal relative direction
* Plane of rotationIn geometry
Geometry (from the grc, γεωμετρία; ''wikt:γῆ, geo-'' "earth", ''wikt:μέτρον, -metron'' "measurement") is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space that ...

* Rotation formalisms in three dimensions
In geometry, various formalisms exist to express a rotation (mathematics), rotation in three dimension (vector space), dimensions as a mathematical transformation (geometry), transformation. In physics, this concept is applied to classical mechanic ...

* Triad Method
Triad is one of the earliest and simplest solutions to the spacecraft attitude determination problem, due to Harold Black. Black played a key role in the development of the guidance, navigation and control of the U.S. Navy's Transit satellite system ...

References

External links

* {{DEFAULTSORT:Orientation (Geometry) Euclidean geometry Rigid bodies Rotation in three dimensions