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Angular displacement of a body is the
angle In Euclidean geometry Euclidean geometry is a mathematical system attributed to Alexandrian Greek mathematics , Greek mathematician Euclid, which he described in his textbook on geometry: the ''Euclid's Elements, Elements''. Euclid's method ...

angle
in
radian The radian, denoted by the symbol \text, is the SI unit The International System of Units, known by the international abbreviation SI in all languages and sometimes pleonastically as the SI system, is the modern form of the metric sy ...

radian
s (
degree Degree may refer to: As a unit of measurement * Degree symbol (°), a notation used in science, engineering, and mathematics * Degree (angle), a unit of angle measurement * Degree (temperature), any of various units of temperature measurement ...
s,
revolutions In political science, a revolution (Latin: ''revolutio'', "a turn around") is a fundamental and relatively sudden change in political power and political organization which occurs when the population revolts against the government, typically due ...
) through which a point revolves around a centre or line has been rotated in a specified sense about a specified
axis Axis may refer to: Politics *Axis of evil The phrase "axis of evil" was first used by U.S. President George W. Bush in his State of the Union address on January 29, 2002, less than five months after the 9/11 attacks, and often repeated t ...

axis
. 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 (''t''). 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. Thus the rotation of a rigid body over a fixed axis is referred to as
rotational motion A rotation is a circular movement of an object around a center (or point) of rotation. The geometric plane along which the rotation occurs is called the '' rotation plane'', and the imaginary line extending from the center and perpendicular ...
.


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 length Arc length is the distance between two points along a section of a curve In mathematics, a curve (also called a curved line in older texts) is an object similar to a line (geometry), line, but that does not have to be Linearity, straight. In ...
''s'', which becomes related to the angular position through the relationship:- :s = r\theta \,


Measurements

Angular displacement may be measured in
radian The radian, denoted by the symbol \text, is the SI unit The International System of Units, known by the international abbreviation SI in all languages and sometimes pleonastically as the SI system, is the modern form of the metric sy ...

radian
s or degrees. Using radians provides a very simple relationship between distance traveled around the circle and the distance ''r'' from the centre. :\theta = \frac 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: \theta= \fracr which easily simplifies to: \theta=2\pi. Therefore, 1 revolution is 2\pi radians. When a particle travels from point P to point Q over \delta t, as it does in the illustration to the left, the radius of the circle goes through a change in angle \Delta \theta = \theta_2 - \theta_1 which equals the ''angular displacement''.


Three dimensions

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 In geometry, Euler's rotation theorem states that, in three-dimensional space, any displacement of a rigid body such that a point on the rigid body remains fixed, is equivalent to a single rotation about some axis that runs through the fixed poi ...
; the magnitude specifies the rotation in
radian The radian, denoted by the symbol \text, is the SI unit The International System of Units, known by the international abbreviation SI in all languages and sometimes pleonastically as the SI system, is the modern form of the metric sy ...

radian
s about that axis (using the
right-hand rule 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 an ...

right-hand rule
to determine direction). This entity is called an axis-angle. Despite having direction and magnitude, angular displacement is not a
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 ...
because it does not obey the
commutative law 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 ge ...
for addition. Nevertheless, when dealing with infinitesimal rotations, second order infinitesimals can be discarded and in this case commutativity appears. Several ways to describe angular displacement exist, like rotation matrices or
Euler angles The Euler angles are three angles introduced by Leonhard Euler Leonhard Euler ( ; ; 15 April 170718 September 1783) was a Swiss mathematician A mathematician is someone who uses an extensive knowledge of mathematics Mathematics ( ...
. See charts on SO(3) for others.


Matrix notation

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 A_0 and A_f two matrices, the angular displacement matrix between them can be obtained as \Delta A = A_f . A_0^. 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

An infinitesimal angular displacement is an
infinitesimal rotation 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 and t ...
matrix: * As any rotation matrix has a single real eigenvalue, which is +1, this eigenvalue shows the rotation axis. * Its module can be deduced from the value of the infinitesimal rotation. * The shape of the matrix is like this: : A = \begin 1 & -d\phi_z(t) & d\phi_y(t) \\ d\phi_z(t) & 1 & -d\phi_x(t) \\ -d\phi_y(t) & d\phi_x(t) & 1 \\ \end We can introduce here the infinitesimal angular displacement tensor or rotation generator associated: : d\Phi(t) = \begin 0 & -d\phi_z(t) & d\phi_y(t) \\ d\phi_z(t) & 0 & -d\phi_x(t) \\ -d\phi_y(t) & d\phi_x(t) & 0 \\ \end Such that its associated rotation matrix is A = I + d\Phi(t). When it is divided by the time, this will yield the
angular velocity 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. ...

angular velocity
vector.


Generators of rotations

Suppose we specify an axis of rotation by a unit vector 'x'', ''y'', ''z''nbsp;, and suppose we have an infinitely small rotation of angle Δθ  about that vector. Expanding the rotation matrix as an infinite addition, and taking the first order approach, the rotation matrix Δ''R''  is represented as: : \Delta R = \begin 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end + \begin 0 & z & -y \\ -z & 0 & x \\ y & -x & 0 \end\,\Delta \theta = \mathbf + \mathbf\,\Delta\theta. A finite rotation through angle θ about this axis may be seen as a succession of small rotations about the same axis. Approximating Δθ  as θ/''N'' where ''N''  is a large number, a rotation of θ about the axis may be represented as: :R = \left(\mathbf + \frac\right)^N \approx e^. It can be seen that Euler's theorem essentially states that all rotations may be represented in this form. The product \mathbf\theta is the "generator" of the particular rotation, being the vector (''x'',''y'',''z'') associated with the matrix A. This shows that the rotation matrix and the axis-angle format are related by the exponential function. One can derive a simple expression for the generator G. One starts with an arbitrary plane defined by a pair of perpendicular unit vectors a and b. In this plane one can choose an arbitrary vector x with perpendicular y. One then solves for y in terms of x and substituting into an expression for a rotation in a plane yields the rotation matrix R which includes the generator G = baT − abT. :\begin x &= a \cos\left( \alpha \right) + b \sin\left( \alpha \right) \\ y &= -a \sin\left( \alpha \right) + b \cos\left( \alpha \right) \\ \cos\left( \alpha \right) &= a^T x \\ \sin\left( \alpha \right) &= b^T x \\ y &= -ab^T x + ba^T x = \left( ba^T - ab^T \right)x \\ \\ x' &= x \cos\left( \beta \right) + y \sin\left( \beta \right) \\ &= \left I \cos\left( \beta \right) + \left( ba^T - ab^T \right) \sin\left( \beta \right) \right \\ \\ R &= I \cos\left( \beta \right) + \left( ba^T - ab^T \right) \sin\left( \beta \right) \\ &= I \cos\left( \beta \right) + G \sin\left( \beta \right) \\ \\ G &= ba^T - ab^T \\ \end To include vectors outside the plane in the rotation one needs to modify the above expression for R by including two projection operators that partition the space. This modified rotation matrix can be rewritten as an
exponential function The exponential function is a mathematical function Function or functionality may refer to: Computing * Function key A function key is a key on a computer A computer is a machine that can be programmed to carry out sequences of ...
. :\begin P_ &= -G^2 \\ R &= I - P_ + \left I \cos\left( \beta \right) + G \sin\left( \beta \right) \rightP_ = e^ \\ \end Analysis is often easier in terms of these generators, rather than the full rotation matrix. Analysis in terms of the generators is known as the
Lie algebra 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 an ...
of the rotation group.


Relationship with Lie algebras

The matrices in the
Lie algebra 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 an ...
are not themselves rotations; the skew-symmetric matrices are derivatives, proportional differences of rotations. An actual "differential rotation", or ''infinitesimal rotation matrix'' has the form : I + A \, d\theta ~, where is vanishingly small and , for instance with , : dL_ = \left begin 1 & 0 & 0 \\ 0 & 1 & -d\theta \\ 0 & d\theta & 1 \end\right The computation rules are as usual except that infinitesimals of second order are routinely dropped. With these rules, these matrices do not satisfy all the same properties as ordinary finite rotation matrices under the usual treatment of infinitesimals. It turns out that ''the order in which infinitesimal rotations are applied is irrelevant''. To see this exemplified, consult infinitesimal rotations SO(3).


Exponential map

Connecting the Lie algebra to the Lie group is the exponential map, which is defined using the standard
matrix exponential 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 ...
series for For any
skew-symmetric matrix In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...
, exp() is always a rotation matrix.Note that this exponential map of skew-symmetric matrices to rotation matrices is quite different from the Cayley transform discussed earlier, differing to 3rd order, e^ - \frac=- \frac A^3 +\mathrm (A^4) ~.
Conversely, a
skew-symmetric matrix In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...
specifying a rotation matrix through the Cayley map specifies the ''same'' rotation matrix through the map .
An important practical example is the case. In
rotation group SO(3) In mechanics Mechanics (Greek Greek may refer to: Greece Anything of, from, or related to Greece Greece ( el, Ελλάδα, , ), officially the Hellenic Republic, is a country located in Southeast Europe. Its population is approximatel ...
, it is shown that one can identify every with an Euler vector , where is a unit magnitude vector. By the properties of the identification , is in the null space of . Thus, is left invariant by and is hence a rotation axis. Using Rodrigues' rotation formula on matrix form with , together with standard double angle formulae one obtains, :\begin \exp( A ) &= \exp(\theta(\boldsymbol)) = \exp \left( \left begin 0 & -z \theta & y \theta \\ z \theta & 0&-x \theta \\ -y \theta & x \theta & 0 \end\right\right)= \boldsymbol + 2\cos\frac\sin\frac~\boldsymbol + 2\sin^2\frac ~(\boldsymbol )^2 ,\end where , . This is the matrix for a rotation around axis by the angle in half-angle form. For full detail, see exponential map SO(3). Notice that for infinitesimal angles second order terms can be ignored and remains exp(A) = I + A


See also

*
Angular distance Angular distance \theta (also known as angular separation, apparent distance, or apparent separation) 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 com ...
*
Angular position Changing orientation of a rotating 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 imagina ...
*
Angular velocity 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. ...

Angular velocity
*
Infinitesimal rotation 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 and t ...
*
Linear elasticity Linear elasticity is a mathematical model of how solid objects deform and become internally stressed due to prescribed loading conditions. It is a simplification of the more general nonlinear theory of elasticity and a branch of continuum mechan ...
*
Second moment of area The second moment of area, or second area moment, or quadratic moment of area and also known as the area moment of inertia, is a geometrical property of an area Area is the quantity that expresses the extent of a two-dimensional region, ...


References

{{Classical mechanics derived SI units Angle