View source for Angular displacement

{{Short description|Displacement measured angle-wise when a body is showing circular or rotational motion}}
{{Infobox physical quantity
| name = Angular displacement
| othernames = rotational displacement, angle of rotation
| width =
| background =
| image = [[File:Examples of Polar Coordinates.svg|class=skin-invert-image|220px]]
| caption = {{longitem|The angle of rotation from the black ray to the green segment is 60°, from the black ray to the blue segment is 210°, and from the green to the blue segment is {{nowrap|1=210° − 60° = 150°}}.  A complete rotation about the center point is equal to 1 [[turn (geometry)|tr]], 360[[degree (angle)|°]], or 2[[pi|π]] [[radian]]s.}}
| unit = [[radians]], [[degrees (angle)|degrees]], [[turn (unit)|turns]], etc. (any [[angular unit]])
| otherunits =
| symbols = θ, {{not a typo|ϑ}}, φ
| baseunits = [[radians]] (rad)
| dimension = 
| extensive =
| intensive =
| conserved =
| transformsas =
| derivations =
}}
{{Classical mechanics|rotational}}

The '''angular displacement''' (symbol θ, {{not a typo|ϑ}}, or φ) – also called '''angle of rotation''', '''rotational displacement''', or '''rotary displacement''' – of a [[physical body]] is the [[angle]] (in [[unit of measurement|units]] of [[radian]]s, [[degree (angle)|degree]]s, [[turn (geometry)|turns]], etc.) through which the body [[rotation|rotates]] (revolves or spins) around a centre or [[axis of rotation]]. Angular displacement may be signed, indicating the sense of rotation (e.g., [[clockwise]]); it may also be greater (in [[absolute value]]) than a full [[turn (angle)|turn]].

==Context==
[[Image:angulardisplacement1.jpg|class=skin-invert-image|upright=1.3|left|thumb|Rotation of a rigid body ''P'' about a fixed axis ''O''.]]
When a body rotates about its axis, the motion cannot simply be analyzed as a particle, as in [[circular motion]] it undergoes a changing velocity and acceleration at any time.  When dealing with the rotation of a body, it becomes simpler to consider the body itself rigid.  A body is generally considered rigid when the separations between all the particles remains constant throughout the body's motion, so for example parts of its mass are not flying off.  In a realistic sense, all things can be deformable, however this impact is minimal and negligible.

== Example ==
In the example illustrated to the right (or above in some mobile versions), a particle or body P is at a fixed distance ''r'' from the origin, ''O'', rotating counterclockwise. It becomes important to then represent the position of particle P in terms of its polar coordinates (''r'', ''θ'').  In this particular example, the value of ''θ'' is changing, while the value of the radius remains the same.  (In rectangular coordinates (''x'', ''y'') both ''x'' and ''y'' vary with time.) As the particle moves along the circle, it travels an [[Arc (geometry)|arc length]] ''s'', which becomes related to the angular position through the relationship:
:<math>s = r\theta .</math>

== Definition and units ==
Angular displacement may be expressed in [[radian]]s or degrees. Using radians provides a very simple relationship between distance traveled around the circle (''[[circular arc]] [[arc length|length]]'') and the distance ''r'' from the centre (''[[radius]]''):

:<math>\theta = \frac{s}{r} \mathrm{rad}</math>

For example, if a body rotates 360° around a circle of radius ''r'', the angular displacement is given by the distance traveled around the circumference - which is 2π''r'' - divided by the radius: <math>\theta= \frac{2\pi r}r</math> which easily simplifies to: <math>\theta=2\pi</math>. Therefore, 1 [[revolution (unit)|revolution]] is <math>2\pi</math> radians.

The above definition is part of the [[International System of Quantities]] (ISQ), formalized in the international standard [[ISO 80000-3]] (Space and time),<ref name="ISO80000-3_2019">{{cite web |title=ISO 80000-3:2019 Quantities and units — Part 3: Space and time |publisher=[[International Organization for Standardization]] |date=2019 |edition=2 |url=https://www.iso.org/standard/64974.html |access-date=2019-10-23}} [https://www.iso.org/obp/ui/#iso:std:iso:80000:-3:ed-2:v1:en] (11 pages)</ref> and adopted in the [[International System of Units]] (SI).<ref name="SIBrochure_9">{{SIbrochure9th}}</ref><ref name="NISTGuide_2009">{{cite web |title=The NIST Guide for the Use of the International System of Units, Special Publication 811 |author-first1=Ambler |author-last1=Thompson |author-first2=Barry N. |author-last2=Taylor |edition=2008 |publisher=[[National Institute of Standards and Technology]] |date=2020-03-04 |orig-date=2009-07-02 |ref={{sfnref|NIST|2009}} |url=https://www.nist.gov/pml/special-publication-811 |access-date=2023-07-17}} [https://web.archive.org/web/20230515201622/https://nvlpubs.nist.gov/nistpubs/Legacy/SP/nistspecialpublication811e2008.pdf]</ref>

Angular displacement may be signed, indicating the sense of rotation (e.g., [[clockwise]]);<ref name="ISO80000-3_2019"/> it may also be greater (in [[absolute value]]) than a full [[turn (angle)|turn]].
In the ISQ/SI, angular displacement is used to define the ''[[number of revolutions]]'', ''N''{{=}}θ/(2π rad), a ratio-type [[quantity of dimension one]].

== In three dimensions ==
{{main|Three-dimensional rotation}}
[[Image:Euler Rotation 2.JPG|200px|left|thumb|'''Figure 1''': Euler's rotation theorem. A great circle transforms to another great circle under rotations, leaving always a diameter of the sphere in its original position.]]
[[Image:Euler AxisAngle.png|thumb|right|'''Figure 2''': A rotation represented by an Euler axis and angle.]]

In three dimensions, angular displacement is an entity with a direction and a magnitude.  The direction specifies the axis of rotation, which always exists by virtue of the [[Euler's rotation theorem]]; the magnitude specifies the rotation in [[radian]]s about that axis (using the [[right-hand rule]] to determine direction). This entity is called an [[axis-angle]].

Despite having direction and magnitude, angular displacement is not a [[vector (geometry)|vector]] because it does not obey the [[commutative law]] for addition.<ref>{{cite book|last1=Kleppner|first1=Daniel|last2=Kolenkow|first2=Robert|title=An Introduction to Mechanics|url=https://archive.org/details/introductiontome00dani|url-access=registration|publisher=McGraw-Hill|year=1973|pages=[https://archive.org/details/introductiontome00dani/page/288 288]–89|isbn=9780070350489}}</ref> Nevertheless, when dealing with infinitesimal rotations, second order infinitesimals can be discarded and in this case commutativity appears.

=== Rotation matrices ===
Several ways to describe rotations exist, like [[rotation matrix|rotation matrices]] or [[Euler angles]]. See [[charts on SO(3)]] for others.

Given that any frame in the space can be described by a rotation matrix, the displacement among them can also be described by a rotation matrix. Being <math>A_0</math> and <math>A_f</math> two matrices, the angular displacement matrix between them can be obtained as <math>\Delta A =  A_f A_0^{-1}</math>. When this product is performed having a very small difference between both frames we will obtain a matrix close to the identity.

In the limit, we will have an infinitesimal rotation matrix.

=== Infinitesimal rotation matrices ===
{{Excerpt|Infinitesimal rotation matrix}}

== See also ==
* [[Angular distance]]
* [[Angular frequency]]
* [[Angular position]]
* [[Angular velocity]]
* [[Azimuth]]
* [[Rotation matrix#Infinitesimal rotations|Infinitesimal rotation]]
* [[Linear elasticity]]
* [[Second moment of area]]
* [[Unwrapped phase]]

== References ==
<references />

=== Sources ===
* {{Citation |last1=Goldstein |first1=Herbert |author1-link=Herbert Goldstein |author2-link=Charles P. Poole |last2=Poole |first2=Charles P. |last3=Safko |first3=John L. |year=2002<!-- January 15 --> |title=Classical Mechanics |edition=third |publisher=[[Addison Wesley]] |isbn=978-0-201-65702-9}}
* {{Citation |last=Wedderburn |first=Joseph H. M. |author-link=Joseph Wedderburn |year=1934 |title=Lectures on Matrices |publisher=[[American Mathematical Society|AMS]] |isbn=978-0-8218-3204-2 |url=https://books.google.com/books?id=6eKVAwAAQBAJ}}

{{Classical mechanics derived SI units}}

[[Category:Angle]]
[[Category:Rotation]]
[[Category:Sign (mathematics)]]