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Euler angles

The Euler angles are three angles introduced by Leonhard Euler to describe the orientation of a rigid body with respect to a fixed coordinate system. They can also represent the orientation of a mobil

3D rotation group

In mechanics and geometry, the 3D rotation group, often denoted SO(3), is the group of all rotations about the origin of three-dimensional Euclidean space under the operation of composition. By defini

Three-dimensional rotation operator

This article derives the main properties of rotations in 3-dimensional space. The three Euler rotations are one way to bring a rigid body to any desired orientation by sequentially making rotations ab

Plane of rotation

In geometry, a plane of rotation is an abstract object used to describe or visualize rotations in space. In three dimensions it is an alternative to the axis of rotation, but unlike the axis of rotati

Rotation formalisms in three dimensions

In geometry, various formalisms exist to express a rotation in three dimensions as a mathematical transformation. In physics, this concept is applied to classical mechanics where rotational (or angula

Spin-1/2

In quantum mechanics, spin is an intrinsic property of all elementary particles. All known fermions, the particles that constitute ordinary matter, have a spin of 1/2. The spin number describes how ma

Nutation

Nutation (from Latin nūtātiō 'nodding, swaying') is a rocking, swaying, or nodding motion in the axis of rotation of a largely axially symmetric object, such as a gyroscope, planet, or bullet in fligh

Rodrigues' rotation formula

In the theory of three-dimensional rotation, Rodrigues' rotation formula, named after Olinde Rodrigues, is an efficient algorithm for rotating a vector in space, given an axis and angle of rotation. B

Clebsch–Gordan coefficients

In physics, the Clebsch–Gordan (CG) coefficients are numbers that arise in angular momentum coupling in quantum mechanics. They appear as the expansion coefficients of total angular momentum eigenstat

Davenport chained rotations

In physics and engineering, Davenport chained rotations are three chained intrinsic rotations about body-fixed specific axes. Euler rotations and Tait–Bryan rotations are particular cases of the Daven

Spinor

In geometry and physics, spinors /spɪnər/ are elements of a complex vector space that can be associated with Euclidean space. Like geometric vectors and more general tensors, spinors transform linearl

Euler's equations (rigid body dynamics)

In classical mechanics, Euler's rotation equations are a vectorial quasilinear first-order ordinary differential equation describing the rotation of a rigid body, using a rotating reference frame with

Plate trick

In mathematics and physics, the plate trick, also known as Dirac's string trick, the belt trick, or the Balinese cup trick, is any of several demonstrations of the idea that rotating an object with st

Quaternion estimator algorithm

The quaternion estimator algorithm (QUEST) is an algorithm designed to solve Wahba's problem, that consists of finding a rotation matrix between two coordinate systems from two sets of observations sa

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 ab

Total angular momentum quantum number

In quantum mechanics, the total angular momentum quantum number parametrises the total angular momentum of a given particle, by combining its orbital angular momentum and its intrinsic angular momentu

Charts on SO(3)

In mathematics, the special orthogonal group in three dimensions, otherwise known as the rotation group SO(3), is a naturally occurring example of a manifold. The various charts on SO(3) set up rival

Chasles' theorem (kinematics)

In kinematics, Chasles' theorem, or Mozzi–Chasles' theorem, says that the most general rigid body displacement can be produced by a translation along a line (called its screw axis or Mozzi axis) follo

Slerp

In computer graphics, Slerp is shorthand for spherical linear interpolation, introduced by Ken Shoemake in the context of quaternion interpolation for the purpose of animating 3D rotation. It refers t

Axis–angle representation

In mathematics, the axis–angle representation of a rotation parameterizes a rotation in a three-dimensional Euclidean space by two quantities: a unit vector e indicating the direction of an axis of ro

Quaternions and spatial rotation

Unit quaternions, known as versors, provide a convenient mathematical notation for representing spatial orientations and rotations of elements in three dimensional space. Specifically, they encode inf

Versor

In mathematics, a versor is a quaternion of norm one (a unit quaternion). The word is derived from Latin versare = "to turn" with the suffix -or forming a noun from the verb (i.e. versor = "the turner

Axes conventions

In ballistics and flight dynamics, axes conventions are standardized ways of establishing the location and orientation of coordinate axes for use as a frame of reference. Mobile objects are normally t

Representation theory of SU(2)

In the study of the representation theory of Lie groups, the study of representations of SU(2) is fundamental to the study of representations of semisimple Lie groups. It is the first case of a Lie gr

Givens rotation

In numerical linear algebra, a Givens rotation is a rotation in the plane spanned by two coordinates axes. Givens rotations are named after Wallace Givens, who introduced them to numerical analysts in

Gimbal lock

Gimbal lock is the loss of one degree of freedom in a three-dimensional, three-gimbal mechanism that occurs when the axes of two of the three gimbals are driven into a parallel configuration, "locking

Triad method

The Triad method is one of the earliest and simplest solutions to the spacecraft attitude determination problem. Given the knowledge of two vectors in the reference and body coordinates of a satellite

Euler–Rodrigues formula

In mathematics and mechanics, the Euler–Rodrigues formula describes the rotation of a vector in three dimensions. It is based on Rodrigues' rotation formula, but uses a different parametrization. The

Orientation (geometry)

In geometry, the orientation, angular position, attitude, bearing, or direction 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

Spin quantum number

In atomic physics, the spin quantum number is a quantum number (designated ms) which describes the intrinsic angular momentum (or spin angular momentum, or simply spin) of an electron or other particl

Conversion between quaternions and Euler angles

Spatial rotations in three dimensions can be parametrized using both Euler angles and unit quaternions. This article explains how to convert between the two representations. Actually this simple use o

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