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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.[1] 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.

Changing orientation of a rigid body is the same as rotating the axes of a reference frame attached to it.

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 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.

Mathematical representations edit

Three dimensions edit

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.

Two dimensions edit

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.

Multiple dimensions edit

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.

Rigid body in three dimensions edit

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

Euler angles edit

 
Euler angles, one of the possible ways to describe an orientation

The first attempt to represent an orientation is attributed to Leonhard Euler. 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.

Tait–Bryan angles edit

 
Tait–Bryan angles. Other way for describing orientation

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.

 
A rotation represented by an Euler axis and angle.

Orientation vector edit

Euler also realized that the composition of two rotations is equivalent to a single rotation about a different fixed axis (Euler's rotation theorem). 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, describes a rotation or orientation using a unit vector aligned with the rotation axis, and a separate value to indicate the angle (see figure).

Orientation matrix edit

With the introduction of matrices, the Euler theorems were rewritten. The rotations were described by orthogonal matrices 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 of a rotation matrix (a rotation matrix has a unique real eigenvalue). 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 object in n-dimensional space is SO(n) × Rn. Orientation may be visualized by attaching a basis of tangent vectors to an object. The direction in which each vector points determines its orientation.

Orientation quaternion edit

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 edit

Miller indices edit

 
Planes with different Miller indices in cubic crystals

The attitude of a lattice plane is the orientation of the line normal to the plane,[2] and is described by the plane's Miller indices. In three-space a family of planes (a series of parallel planes) can be denoted by its Miller indices (hkl),[3][4] so the family of planes has an attitude common to all its constituent planes.

Strike and dip edit

 
Strike line and dip of a plane describing attitude relative to a horizontal plane and a vertical plane perpendicular to the strike line

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.[5]

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 geographic north or from magnetic north). 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 edit

Rigid body edit

 
The orientation of a rigid body is determined by three angles

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.[6] 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 Euler angles.[7][8] Another is based upon roll, pitch and yaw,[9] although these terms also refer to incremental deviations from the nominal attitude

See also edit

References edit

  1. ^ Robert J. Twiss; Eldridge M. Moores (1992). "§2.1 The orientation of structures". Structural Geology (2nd ed.). Macmillan. p. 11. ISBN 0-7167-2252-6. ...the attitude of a plane or a line — that is, its orientation in space — is fundamental to the description of structures.
  2. ^ William Anthony Granville (1904). "§178 Normal line to a surface". Elements of the Differential and Integral Calculus. Ginn & Company. p. 275.
  3. ^ Augustus Edward Hough Love (1892). A Treatise on the Mathematical Theory of Elasticity. Vol. 1. Cambridge University Press. p. 79 ff.
  4. ^ Marcus Frederick Charles Ladd; Rex Alfred Palmer (2003). "§2.3 Families of planes and interplanar spacings". Structure Determination by X-Ray Crystallography (4th ed.). Springer. p. 62 ff. ISBN 0-306-47454-9.
  5. ^ Stephen Mark Rowland; Ernest M. Duebendorfer; Ilsa M. Schiefelbein (2007). "Attitudes of lines and planes". Structural Analysis and Synthesis: A Laboratory Course in Structural Geology (3rd ed.). Wiley-Blackwell. p. 1 ff. ISBN 978-1-4051-1652-7.
  6. ^ Hanspeter Schaub; John L. Junkins (2003). "Rigid body kinematics". Analytical Mechanics of Space Systems. American Institute of Aeronautics and Astronautics. p. 71. ISBN 1-56347-563-4.
  7. ^ Jack B. Kuipers (2002). "Figure 4.7: Aircraft Euler angle sequence". Quaternions and Rotation Sequences: A Primer with Applications to Orbits, Aerospace, and Virtual Reality. Princeton University Press. p. 85. ISBN 0-691-10298-8.
  8. ^ Bong Wie (1998). "§5.2 Euler angles". Space Vehicle Dynamics and Control. American Institute of Aeronautics and Astronautics. p. 310. ISBN 1-56347-261-9. Euler angle rigid body attitude.
  9. ^ Lorenzo Sciavicco; Bruno Siciliano (2000). "§2.4.2 Roll–pitch–yaw angles". Modelling and Control of Robot Manipulators (2nd ed.). Springer. p. 32. ISBN 1-85233-221-2.

External links edit

  •   Media related to Orientation (mathematics) at Wikimedia Commons

orientation, geometry, this, article, about, orientation, attitude, object, shape, space, orientation, space, orientability, geometry, orientation, attitude, bearing, direction, angular, position, object, such, line, plane, rigid, body, part, description, plac. This article is about the orientation or attitude of an object or a shape in a space For the orientation of a space see Orientability 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 1 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 Changing orientation of a rigid body is the same as rotating the axes of a reference frame attached to it 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 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 Contents 1 Mathematical representations 1 1 Three dimensions 1 2 Two dimensions 1 3 Multiple dimensions 2 Rigid body in three dimensions 2 1 Euler angles 2 1 1 Tait Bryan angles 2 2 Orientation vector 2 3 Orientation matrix 2 4 Orientation quaternion 3 Plane in three dimensions 3 1 Miller indices 3 2 Strike and dip 4 Usage examples 4 1 Rigid body 5 See also 6 References 7 External linksMathematical representations editThree dimensions edit 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 Two dimensions edit 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 Multiple dimensions edit 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 Rigid body in three dimensions editMain article Rotation formalisms 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 edit Main article Euler angles nbsp Euler angles one of the possible ways to describe an orientationThe first attempt to represent an orientation is attributed to Leonhard Euler 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 Tait Bryan angles edit Main article Euler angles Tait Bryan angles nbsp Tait Bryan angles Other way for describing orientationThese 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 nbsp A rotation represented by an Euler axis and angle Orientation vector edit Main article Axis angle representation Euler also realized that the composition of two rotations is equivalent to a single rotation about a different fixed axis Euler s rotation theorem 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 describes a rotation or orientation using a unit vector aligned with the rotation axis and a separate value to indicate the angle see figure Orientation matrix edit Main article Rotation matrix With the introduction of matrices the Euler theorems were rewritten The rotations were described by orthogonal matrices 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 of a rotation matrix a rotation matrix has a unique real eigenvalue 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 object in n dimensional space is SO n Rn Orientation may be visualized by attaching a basis of tangent vectors to an object The direction in which each vector points determines its orientation Orientation quaternion edit Main article Quaternions and spatial rotation 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 editMiller indices edit Main article Miller index nbsp Planes with different Miller indices in cubic crystalsThe attitude of a lattice plane is the orientation of the line normal to the plane 2 and is described by the plane s Miller indices In three space a family of planes a series of parallel planes can be denoted by its Miller indices hkl 3 4 so the family of planes has an attitude common to all its constituent planes Strike and dip edit Main article Strike and dip nbsp Strike line and dip of a plane describing attitude relative to a horizontal plane and a vertical plane perpendicular to the strike lineMany 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 5 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 geographic north or from magnetic north 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 editRigid body edit Main article Rigid body nbsp The orientation of a rigid body is determined by three anglesThe 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 6 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 Euler angles 7 8 Another is based upon roll pitch and yaw 9 although these terms also refer to incremental deviations from the nominal attitudeSee also editAngular displacement Attitude control Body relative direction Directional statistics Oriented area Plane of rotation Rotation formalisms in three dimensions Signed direction Terms of orientation Triad methodReferences edit Robert J Twiss Eldridge M Moores 1992 2 1 The orientation of structures Structural Geology 2nd ed Macmillan p 11 ISBN 0 7167 2252 6 the attitude of a plane or a line that is its orientation in space is fundamental to the description of structures William Anthony Granville 1904 178 Normal line to a surface Elements of the Differential and Integral Calculus Ginn amp Company p 275 Augustus Edward Hough Love 1892 A Treatise on the Mathematical Theory of Elasticity Vol 1 Cambridge University Press p 79 ff Marcus Frederick Charles Ladd Rex Alfred Palmer 2003 2 3 Families of planes and interplanar spacings Structure Determination by X Ray Crystallography 4th ed Springer p 62 ff ISBN 0 306 47454 9 Stephen Mark Rowland Ernest M Duebendorfer Ilsa M Schiefelbein 2007 Attitudes of lines and planes Structural Analysis and Synthesis A Laboratory Course in Structural Geology 3rd ed Wiley Blackwell p 1 ff ISBN 978 1 4051 1652 7 Hanspeter Schaub John L Junkins 2003 Rigid body kinematics Analytical Mechanics of Space Systems American Institute of Aeronautics and Astronautics p 71 ISBN 1 56347 563 4 Jack B Kuipers 2002 Figure 4 7 Aircraft Euler angle sequence Quaternions and Rotation Sequences A Primer with Applications to Orbits Aerospace and Virtual Reality Princeton University Press p 85 ISBN 0 691 10298 8 Bong Wie 1998 5 2 Euler angles Space Vehicle Dynamics and Control American Institute of Aeronautics and Astronautics p 310 ISBN 1 56347 261 9 Euler angle rigid body attitude Lorenzo Sciavicco Bruno Siciliano 2000 2 4 2 Roll pitch yaw angles Modelling and Control of Robot Manipulators 2nd ed Springer p 32 ISBN 1 85233 221 2 External links edit nbsp Media related to Orientation mathematics at Wikimedia Commons Retrieved from https en wikipedia org w index php title Orientation geometry amp oldid 1217992214 Plane in three dimensions, wikipedia, wiki, book, books, library,

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