This article derives the main properties of rotations in 3-dimensional space.
Rotation in mathematics is a concept originating in geometry. Any rotation is a motion of a certain space that preserves at least one point. It can describe, for example, the motion of a rigid body around a fixed point. A rotation is different from other types of motions: translations, which have no fixed points, and (hyperplane) reflections, each of them having an entire (n − 1)-dimensional flat of fixed points in a n-dimensional space. A clockwise rotation is a negative magnitude so a counterclockwise turn has a positive magnitude.
The three Euler rotations are one way to bring a rigid body to any desired orientation by sequentially making rotations about axis' fixed relative to the object. However, this can also be achieved with one single rotation (Euler's rotation theorem). Using the concepts of linear algebra it is shown how this single rotation can be performed.
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 mobile frame of reference in physics or the orientation of a general basis in 3-dimensional linear algebra.
In physics, a rigid body is a solid body in which deformation is zero or so small it can be neglected. The distance between any two given points on a rigid body remains constant in time regardless of external forces exerted on it. A rigid body is usually considered as a continuous distribution of mass.
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 point. It also means that the composition of two rotations is also a rotation. Therefore the set of rotations has a group structure, known as a rotation group.
be a coordinate system fixed in the body that through a change in orientation is brought to the new directions
In geometry, a coordinate system is a system that uses one or more numbers, or coordinates, to uniquely determine the position of the points or other geometric elements on a manifold such as Euclidean space. The order of the coordinates is significant, and they are sometimes identified by their position in an ordered tuple and sometimes by a letter, as in "the x-coordinate". The coordinates are taken to be real numbers in elementary mathematics, but may be complex numbers or elements of a more abstract system such as a commutative ring. The use of a coordinate system allows problems in geometry to be translated into problems about numbers and vice versa; this is the basis of analytic geometry.
rotating with the body is then brought to the new direction
i.e. this is a linear operator
The matrix of this operator relative to the coordinate system
In mathematics, a matrix is a rectangular array of numbers, symbols, or expressions, arranged in rows and columns. For example, the dimensions of the matrix below are 2 × 3, because there are two rows and three columns:
In mathematics, an operator is generally a mapping that acts on elements of a space to produce other elements of the same space. The most common operators are linear maps, which act on vector spaces. However, when using "linear operator" instead of "linear map", mathematicians often mean actions on vector spaces of functions, which also preserve other properties, such as continuity. For example, differentiation and indefinite integration are linear operators; operators that are built from them are called differential operators, integral operators or integro-differential operators.
or equivalently in matrix notation
the matrix is orthogonal and as a "right hand" base vector system is re-orientated into another "right hand" system the determinant of this matrix has the value 1.
An orthogonal matrix is a square matrix whose columns and rows are orthogonal unit vectors, i.e.
In linear algebra, the determinant is a value that can be computed from the elements of a square matrix and encodes certain properties of the linear transformation described by the matrix. The determinant of a matrix A is denoted det(A), det A, or |A|. Geometrically, it can be viewed as the volume scaling factor of the linear transformation described by the matrix. This is also the signed volume of the n-dimensional parallelepiped spanned by the column or row vectors of the matrix. The determinant is positive or negative according to whether the linear mapping preserves or reverses the orientation of n-space.
be an orthogonal positively oriented base vector system in .
The linear operator
"Rotation with the angle around the axis defined by "
has the matrix representation
relative to this basevector system.
This then means that a vector
is rotated to the vector
by the linear operator.
The determinant of this matrix is
and the characteristic polynomial is
The matrix is symmetric if and only if , i.e. for and for .
The case is the trivial case of an identity operator.
For the case the characteristic polynomial is
In linear algebra, the characteristic polynomial of a square matrix is a polynomial which is invariant under matrix similarity and has the eigenvalues as roots. It has the determinant and the trace of the matrix as coefficients. The characteristic polynomial of an endomorphism of vector spaces of finite dimension is the characteristic polynomial of the matrix of the endomorphism over any base; it does not depend on the choice of a basis. The characteristic equation is the equation obtained by equating to zero the characteristic polynomial.
i.e. the rotation operator has the eigenvalues
The eigenspace corresponding to is all vectors on the rotation axis, i.e. all vectors
The eigenspace corresponding to consists of all vectors orthogonal to the rotation axis, i.e. all vectors
For all other values of the matrix is un-symmetric and as there is only the eigenvalue with the one-dimensional eigenspace of the vectors on the rotation axis:
The rotation matrix by angle around a general axis of rotation is given by Rodrigues' rotation formula.
where is the identity matrix and is the dual 2-form of or cross product matrix,
Note that satisfies for all .
"Rotation with the angle around a specified axis"
discussed above is an orthogonal mapping and its matrix relative to any base vector system is therefore an orthogonal matrix . Furthermore its determinant has the value 1. A non-trivial fact is the opposite, i.e. that for any orthogonal linear mapping in having determinant = 1 there exist base vectors
such that the matrix takes the "canonical form"
for some value of .
In fact, if a linear operator has the orthogonal matrix
relative some base vector system
and this matrix is symmetric, the "Symmetric operator theorem" valid in (any dimension) applies saying
that it has n orthogonal eigenvectors. This means for the 3-dimensional case that there exists a coordinate system
such that the matrix takes the form
As it is an orthogonal matrix these diagonal elements are either 1 or −1. As the determinant is 1 these elements are either all 1 or one of the elements is 1 and the other two are −1.
In the first case it is the trivial identity operator corresponding to .
In the second case it has the form
if the basevectors are numbered such that the one with eigenvalue 1 has index 3. This matrix is then of the desired form for .
If the matrix is un-symmetric, the vector
is non-zero. This vector is an eigenvector with eigenvalue
and selecting any two orthogonal unit vectors in the plane orthogonal to :
form a positively oriented triple, the operator takes the desired form with
The expressions above are in fact valid also for the case of a symmetric rotation operator corresponding to a rotation with or . But the difference is that for the vector
is zero and of no use for finding the eigenspace of eigenvalue 1, i.e. the rotation axis.
Defining as the matrix for the rotation operator is
i.e. except for the cases (the identity operator) and
Quaternions are defined similar to with the difference that the half angle is used instead of the full angle .
This means that the first 3 components are components of a vector defined from
and that the fourth component is the scalar
As the angle defined from the canonical form is in the interval
one would normally have that . But a "dual" representation of a rotation with quaternions is used, i.e.
are two alternative representations of one and the same rotation.
The entities are defined from the quaternions by
Using quaternions the matrix of the rotation operator is
Consider the reorientation corresponding to the Euler angles relative a given base vector system
Corresponding matrix relative to this base vector system is (see Euler angles#Matrix orientation)
and the quaternion is
The canonical form of this operator
with is obtained with
The quaternion relative to this new system is then
Instead of making the three Euler rotations
the same orientation can be reached with one single rotation of size around
In physics, the Lorentz transformations are a one-parameter family of linear transformations from a coordinate frame in space time to another frame that moves at a constant velocity, the parameter, within the former. The transformations are named after the Dutch physicist Hendrik Lorentz. The respective inverse transformation is then parametrized by the negative of this velocity.
Kinematics is a branch of classical mechanics that describes the motion of points, bodies (objects), and systems of bodies without considering the forces that caused the motion. Kinematics, as a field of study, is often referred to as the "geometry of motion" and is occasionally seen as a branch of mathematics. A kinematics problem begins by describing the geometry of the system and declaring the initial conditions of any known values of position, velocity and/or acceleration of points within the system. Then, using arguments from geometry, the position, velocity and acceleration of any unknown parts of the system can be determined. The study of how forces act on bodies falls within kinetics, not kinematics. For further details, see analytical dynamics.
Angular displacement of a body is the angle in radians through which a point revolves around a centre or line has been rotated in a specified sense about a specified 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.
In mathematics, a unit vector in a normed vector space is a vector of length 1. A unit vector is often denoted by a lowercase letter with a circumflex, or "hat": . The term direction vector is used to describe a unit vector being used to represent spatial direction, and such quantities are commonly denoted as d. Two 2D direction vectors, d1 and d2 are illustrated. 2D spatial directions represented this way are numerically equivalent to points on the unit circle.
Ray transfer matrix analysis is a type of ray tracing technique used in the design of some optical systems, particularly lasers. It involves the construction of a ray transfer matrix which describes the optical system; tracing of a light path through the system can then be performed by multiplying this matrix with a vector representing the light ray. The same analysis is also used in accelerator physics to track particles through the magnet installations of a particle accelerator, see beam optics.
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 R3 under the operation of composition. By definition, a rotation about the origin is a transformation that preserves the origin, Euclidean distance, and orientation. Every non-trivial rotation is determined by its axis of rotation and its angle of rotation. Composing two rotations results in another rotation; every rotation has a unique inverse rotation; and the identity map satisfies the definition of a rotation. Owing to the above properties, the set of all rotations is a group under composition. Rotations are not commutative, making it a nonabelian group. Moreover, the rotation group has a natural structure as a manifold for which the group operations are smoothly differentiable; so it is in fact a Lie group. It is compact and has dimension 3.
In mathematics, particularly in linear algebra, a skew-symmetricmatrix is a square matrix whose transpose equals its negative, that is, it satisfies the condition
Unit quaternions, also known as versors, provide a convenient mathematical notation for representing orientations and rotations of objects in three dimensions. Compared to Euler angles they are simpler to compose and avoid the problem of gimbal lock. Compared to rotation matrices they are more compact, more numerically stable, and more efficient. Quaternions have applications in computer graphics, computer vision, robotics, navigation, molecular dynamics, flight dynamics, orbital mechanics of satellites and crystallographic texture analysis.
In vector calculus, the Jacobian matrix is the matrix of all first-order partial derivatives of a vector-valued function. When the matrix is a square matrix, both the matrix and its determinant are referred to as the Jacobian in literature.
In the mathematical field of differential geometry, a metric tensor is a type of function which takes as input a pair of tangent vectors v and w at a point of a surface and produces a real number scalar g(v, w) in a way that generalizes many of the familiar properties of the dot product of vectors in Euclidean space. In the same way as a dot product, metric tensors are used to define the length of and angle between tangent vectors. Through integration, the metric tensor allows one to define and compute the length of curves on the manifold.
In linear algebra, linear transformations can be represented by matrices. If is a linear transformation mapping to and is a column vector with entries, then
In linear algebra, a rotation matrix is a matrix that is used to perform a rotation in Euclidean space. For example, using the convention below, the matrix
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 all three basis vectors to compute a rotation matrix in SO(3), the group of all rotation matrices, from an axis–angle representation. In other words, the Rodrigues' formula provides an algorithm to compute the exponential map from so(3), the Lie algebra of SO(3), to SO(3) without actually computing the full matrix exponential.
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 kinematics is the science of quantitative description of a purely rotational motion. The orientation of an object at a given instant is described with the same tools, as it is defined as an imaginary rotation from a reference placement in space, rather than an actually observed rotation from a previous placement in space.
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 rotation, and an angle θ describing the magnitude of the rotation about the axis. Only two numbers, not three, are needed to define the direction of a unit vector e rooted at the origin because the magnitude of e is constrained. For example, the elevation and azimuth angles of e suffice to locate it in any particular Cartesian coordinate frame.
Deformation in continuum mechanics is the transformation of a body from a reference configuration to a current configuration. A configuration is a set containing the positions of all particles of the body.
In fluid dynamics, the Oseen equations describe the flow of a viscous and incompressible fluid at small Reynolds numbers, as formulated by Carl Wilhelm Oseen in 1910. Oseen flow is an improved description of these flows, as compared to Stokes flow, with the (partial) inclusion of convective acceleration.
There are common integrals in quantum field theory that appear repeatedly. These integrals are all variations and generalizations of gaussian integrals to the complex plane and to multiple dimensions. Other integrals can be approximated by versions of the gaussian integral. Fourier integrals are also considered.
The direct-quadrature-zerotransformation or zero-direct-quadraturetransformation is a tensor that rotates the reference frame of a three-element vector or a three-by-three element matrix in an effort to simplify analysis. The DQZ transform is the product of the Clarke transform and the Park transform, first proposed in 1929 by Robert H. Park.