The angular displacement (symbol θ, ϑ, or φ) – also called angle of rotation, rotational displacement, or rotary displacement – of a physical body is the angle (in units of radians, degrees, turns, etc.)

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.

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.

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.

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).

As the particle moves along the circle, it travels an arc length s, which becomes related to the angular position through the relationship: Angular displacement may be expressed in radians or degrees.

Using radians provides a very simple relationship between distance traveled around the circle (circular arc length) and the distance r from the centre (radius): 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 2πr - divided by the radius:

[2][3] Angular displacement may be signed, indicating the sense of rotation (e.g., clockwise);[1] it may also be greater (in absolute value) than a full 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, angular displacement is an entity with a direction and a magnitude.

Despite having direction and magnitude, angular displacement is not a vector because it does not obey the commutative law for addition.

When this product is performed having a very small difference between both frames we will obtain a matrix close to the identity.

(the special orthogonal group), the differential of a rotation is a skew-symmetric matrix

(the special orthogonal Lie algebra), which is not itself a rotation matrix.

representing an infinitesimal three-dimensional rotation about the x-axis, a basis element of