In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
and
mechanics
Mechanics (from Ancient Greek: μηχανική, ''mēkhanikḗ'', "of machines") is the area of mathematics and physics concerned with the relationships between force, matter, and motion among physical objects. Forces applied to objects r ...
, the Euler–Rodrigues formula describes the rotation of a vector in three dimensions. It is based on
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. By extension, this can be used to transform al ...
, but uses a different parametrization.
The rotation is described by four Euler parameters due to
Leonhard Euler
Leonhard Euler ( , ; 15 April 170718 September 1783) was a Swiss mathematician, physicist, astronomer, geographer, logician and engineer who founded the studies of graph theory and topology and made pioneering and influential discoveries in ma ...
. The Rodrigues formula (named after
Olinde Rodrigues
Benjamin Olinde Rodrigues (6 October 1795 – 17 December 1851), more commonly known as Olinde Rodrigues, was a French banker, mathematician, and social reformer. In mathematics Rodrigues is remembered for Rodrigues' rotation formula for vectors, ...
), a method of calculating the position of a rotated point, is used in some software applications, such as
flight simulator
A flight simulator is a device that artificially re-creates aircraft flight and the environment in which it flies, for pilot training, design, or other purposes. It includes replicating the equations that govern how aircraft fly, how they rea ...
s and
computer games
A personal computer game, also known as a PC game or computer game, is a type of video game played on a personal computer (PC) rather than a video game console or arcade machine. Its defining characteristics include: more diverse and user-deter ...
.
Definition
A rotation about the origin is represented by four real numbers, , , , such that
:
When the rotation is applied, a point at position rotates to its new position
:
Vector formulation
The parameter may be called the ''scalar'' parameter and the ''vector'' parameter. In standard vector notation, the Rodrigues rotation formula takes the compact form
Symmetry
The parameters and describe the same rotation. Apart from this symmetry, every set of four parameters describes a unique rotation in three-dimensional space.
Composition of rotations
The composition of two rotations is itself a rotation. Let and be the Euler parameters of two rotations. The parameters for the compound rotation (rotation 2 after rotation 1) are as follows:
:
It is straightforward, though tedious, to check that . (This is essentially
Euler's four-square identity
In mathematics, Euler's four-square identity says that the product of two numbers, each of which is a sum of four square (algebra), squares, is itself a sum of four squares.
Algebraic identity
For any pair of quadruples from a commutative ring, th ...
, also used by Rodrigues.)
Rotation angle and rotation axis
Any central rotation in three dimensions is uniquely determined by its axis of rotation (represented by a
unit vector
In mathematics, a unit vector in a normed vector space is a vector (often a spatial vector) of length 1. A unit vector is often denoted by a lowercase letter with a circumflex, or "hat", as in \hat (pronounced "v-hat").
The term ''direction vecto ...
) and the rotation angle . The Euler parameters for this rotation are calculated as follows:
:
Note that if is increased by a full rotation of 360 degrees, the arguments of sine and cosine only increase by 180 degrees. The resulting parameters are the opposite of the original values, ; they represent the same rotation.
In particular, the identity transformation (null rotation, ) corresponds to parameter values . Rotations of 180 degrees about any axis result in .
Connection with quaternions
The Euler parameters can be viewed as the coefficients of a
quaternion
In mathematics, the quaternion number system extends the complex numbers. Quaternions were first described by the Irish mathematician William Rowan Hamilton in 1843 and applied to mechanics in three-dimensional space. Hamilton defined a quatern ...
; the scalar parameter is the real part, the vector parameters , , are the imaginary parts.
Thus we have the quaternion
:
which is a quaternion of unit length (or
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"). It was introduced by Willi ...
) since
:
Most importantly, the above equations for composition of rotations are precisely the equations for multiplication of quaternions. In other words, the group of unit quaternions with multiplication, modulo the negative sign, is isomorphic to the group of rotations with composition.
Connection with SU(2) spin matrices
The
Lie group
In mathematics, a Lie group (pronounced ) is a group that is also a differentiable manifold. A manifold is a space that locally resembles Euclidean space, whereas groups define the abstract concept of a binary operation along with the additio ...
SU(2)
In mathematics, the special unitary group of degree , denoted , is the Lie group of unitary matrices with determinant 1.
The more general unitary matrices may have complex determinants with absolute value 1, rather than real 1 in the special ...
can be used to represent three-dimensional rotations in complex matrices. The SU(2)-matrix corresponding to a rotation, in terms of its Euler parameters, is
:
Alternatively, this can be written as the sum
:
where the are the
Pauli spin matrices
In mathematical physics and mathematics, the Pauli matrices are a set of three complex matrices which are Hermitian, involutory and unitary. Usually indicated by the Greek letter sigma (), they are occasionally denoted by tau () when used in ...
. Thus, the Euler parameters are the real and imaginary coordinates in an SU(2) matrix corresponding to an element of the
spin group
In mathematics the spin group Spin(''n'') page 15 is the double cover of the special orthogonal group , such that there exists a short exact sequence of Lie groups (when )
:1 \to \mathrm_2 \to \operatorname(n) \to \operatorname(n) \to 1.
As a L ...
Spin(3), which maps by a double cover mapping to a rotation in the
orthogonal group
In mathematics, the orthogonal group in dimension , denoted , is the Group (mathematics), group of isometry, distance-preserving transformations of a Euclidean space of dimension that preserve a fixed point, where the group operation is given by ...
SO(3). This realizes
as the unique three-dimensional irreducible
representation of the
Lie group
In mathematics, a Lie group (pronounced ) is a group that is also a differentiable manifold. A manifold is a space that locally resembles Euclidean space, whereas groups define the abstract concept of a binary operation along with the additio ...
SU(2) ≈ Spin(3).
See also
*
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 angular) kinematics is the science of quantitative de ...
*
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 information about an axis-angle rotation abou ...
*
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"). It was introduced by Willi ...
*
Spinors in three dimensions
In mathematics, the spinor concept as specialised to three dimensions can be treated by means of the traditional notions of dot product and cross product. This is part of the detailed algebraic discussion of the rotation group SO(3).
Formulation
T ...
*
SO(4)
In mathematics, the group of rotations about a fixed point in four-dimensional Euclidean space is denoted SO(4). The name comes from the fact that it is the special orthogonal group of order 4.
In this article ''rotation'' means ''rotational dis ...
*
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 \R^3 under the operation of composition.
By definition, a rotation about the origin is a tr ...
References
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{{DEFAULTSORT:Euler-Rodrigues Parameters
Rotation in three dimensions
Euclidean symmetries
Leonhard Euler