Editing Angular displacement (section)

== 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}}