Rotation 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 ...
is a concept originating in
geometry
Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is c ...
. Any rotation is a
motion
In physics, motion is the phenomenon in which an object changes its position with respect to time. Motion is mathematically described in terms of displacement, distance, velocity, acceleration, speed and frame of reference to an observer and m ...
of a certain
space
Space is the boundless three-dimensional extent in which objects and events have relative position and direction. In classical physics, physical space is often conceived in three linear dimensions, although modern physicists usually consi ...
that preserves at least one
point. It can describe, for example, the motion of a
rigid body around a fixed point. Rotation can have
sign (as in the
sign of an angle): a clockwise rotation is a negative magnitude so a counterclockwise turn has a positive magnitude.
A rotation is different from other types of motions:
translations, which have no fixed points, and
(hyperplane) reflections, each of them having an entire -dimensional
flat of fixed points in a -
dimensional
In physics and mathematics, the dimension of a mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any point within it. Thus, a line has a dimension of one (1D) because only one coordi ...
space.
Mathematically, a rotation is a
map. All rotations about a fixed point form a
group under
composition called the rotation group (of a particular space). But in
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 objec ...
and, more generally, in
physics
Physics is the natural science that studies matter, its fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge which ...
, this concept is frequently understood as a
coordinate transformation (importantly, a transformation of an
orthonormal basis
In mathematics, particularly linear algebra, an orthonormal basis for an inner product space ''V'' with finite dimension is a basis for V whose vectors are orthonormal, that is, they are all unit vectors and orthogonal to each other. For examp ...
), because for any motion of a body there is an inverse transformation which if applied to the
frame of reference
In physics and astronomy, a frame of reference (or reference frame) is an abstract coordinate system whose origin, orientation, and scale are specified by a set of reference points― geometric points whose position is identified both math ...
results in the body being at the same coordinates. For example, in two dimensions rotating a body
clockwise
Two-dimensional rotation can occur in two possible directions. Clockwise motion (abbreviated CW) proceeds in the same direction as a clock's hands: from the top to the right, then down and then to the left, and back up to the top. The opposite ...
about a point keeping the axes fixed is equivalent to rotating the axes counterclockwise about the same point while the body is kept fixed. These two types of rotation are called
active and passive transformations.
Related definitions and terminology
The ''rotation group'' is a
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 addi ...
of rotations about a
fixed point. This (common) fixed point is called the ''
center of rotation'' and is usually identified with the
origin
Origin(s) or The Origin may refer to:
Arts, entertainment, and media
Comics and manga
* Origin (comics), ''Origin'' (comics), a Wolverine comic book mini-series published by Marvel Comics in 2002
* The Origin (Buffy comic), ''The Origin'' (Bu ...
. The rotation group is a ''
point stabilizer'' in a broader group of (orientation-preserving)
motions.
For a particular rotation:
* The ''axis of rotation'' is a
line of its fixed points. They exist only in .
* The ''
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 rotation it can be used in other dimensions, such as ...
'' is a
plane that is
invariant under the rotation. Unlike the axis, its points are not fixed themselves. The axis (where present) and the plane of a rotation are
orthogonal.
A ''representation'' of rotations is a particular formalism, either algebraic or geometric, used to parametrize a rotation map. This meaning is somehow inverse to
the meaning in the group theory.
Rotations of
(affine) spaces of points and of respective
vector space
In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called '' vectors'', may be added together and multiplied ("scaled") by numbers called ''scalars''. Scalars are often real numbers, but can ...
s are not always clearly distinguished. The former are sometimes referred to as ''affine rotations'' (although the term is misleading), whereas the latter are ''vector rotations''. See the article below for details.
Definitions and representations
In Euclidean geometry
A motion of a
Euclidean space
Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidea ...
is the same as its
isometry
In mathematics, an isometry (or congruence, or congruent transformation) is a distance-preserving transformation between metric spaces, usually assumed to be bijective. The word isometry is derived from the Ancient Greek: ἴσος ''isos'' ...
: it leaves
the distance between any two points unchanged after the transformation. But a (proper) rotation also has to preserve the
orientation structure. The "
improper rotation" term refers to isometries that reverse (flip) the orientation. In the language of
group theory
In abstract algebra, group theory studies the algebraic structures known as groups.
The concept of a group is central to abstract algebra: other well-known algebraic structures, such as rings, fields, and vector spaces, can all be seen ...
the distinction is expressed as ''direct'' vs ''indirect'' isometries in the
Euclidean group, where the former comprise the
identity component. Any direct Euclidean motion can be represented as a composition of a rotation about the fixed point and a translation.
There are no non-
trivial
Trivia is information and data that are considered to be of little value. It can be contrasted with general knowledge and common sense.
Latin Etymology
The ancient Romans used the word ''triviae'' to describe where one road split or fork ...
rotations in one dimension. In
two dimensions, only a single
angle
In Euclidean geometry, an angle is the figure formed by two rays, called the '' sides'' of the angle, sharing a common endpoint, called the ''vertex'' of the angle.
Angles formed by two rays lie in the plane that contains the rays. Angles ...
is needed to specify a rotation about the
origin
Origin(s) or The Origin may refer to:
Arts, entertainment, and media
Comics and manga
* Origin (comics), ''Origin'' (comics), a Wolverine comic book mini-series published by Marvel Comics in 2002
* The Origin (Buffy comic), ''The Origin'' (Bu ...
– the ''angle of rotation'' that specifies an element of the
circle group (also known as ). The rotation is acting to rotate an object
counterclockwise
Two-dimensional rotation can occur in two possible directions. Clockwise motion (abbreviated CW) proceeds in the same direction as a clock's hands: from the top to the right, then down and then to the left, and back up to the top. The opposite ...
through an angle about the
origin
Origin(s) or The Origin may refer to:
Arts, entertainment, and media
Comics and manga
* Origin (comics), ''Origin'' (comics), a Wolverine comic book mini-series published by Marvel Comics in 2002
* The Origin (Buffy comic), ''The Origin'' (Bu ...
; see
below
Below may refer to:
*Earth
* Ground (disambiguation)
*Soil
*Floor
* Bottom (disambiguation)
*Less than
*Temperatures below freezing
*Hell or underworld
People with the surname
*Ernst von Below (1863–1955), German World War I general
*Fred Below ...
for details. Composition of rotations
sums their angles
modulo
In computing, the modulo operation returns the remainder or signed remainder of a division, after one number is divided by another (called the '' modulus'' of the operation).
Given two positive numbers and , modulo (often abbreviated as ) is ...
1
turn, which implies that all two-dimensional rotations about ''the same'' point
commute
Commute, commutation or commutative may refer to:
* Commuting, the process of travelling between a place of residence and a place of work
Mathematics
* Commutative property, a property of a mathematical operation whose result is insensitive to th ...
. Rotations about ''different'' points, in general, do not commute. Any two-dimensional direct motion is either a translation or a rotation; see
Euclidean plane isometry for details.
Rotations in
three-dimensional space
Three-dimensional space (also: 3D space, 3-space or, rarely, tri-dimensional space) is a geometric setting in which three values (called ''parameters'') are required to determine the position of an element (i.e., point). This is the informa ...
differ from those in two dimensions in a number of important ways. Rotations in three dimensions are generally not
commutative, so the order in which rotations are applied is important even about the same point. Also, unlike the two-dimensional case, a three-dimensional direct motion, in
general position, is not a rotation but a
screw operation. Rotations about the origin have three degrees of freedom (see
rotation formalisms in three dimensions for details), the same as the number of dimensions.
A three-dimensional rotation can be specified in a number of ways. The most usual methods are:
*
Euler angles (pictured at the left). Any rotation about the origin can be represented as the
composition of three rotations defined as the motion obtained by changing one of the Euler angles while leaving the other two constant. They constitute a mixed axes of rotation system because angles are measured with respect to a mix of different
reference frames, rather than a single frame that is purely external or purely intrinsic. Specifically, the first angle moves the
line of nodes around the external axis ''z'', the second rotates around the line of nodes and the third is an intrinsic rotation (a spin) around an axis fixed in the body that moves. Euler angles are typically denoted as
''α'',
''β'',
''γ'', or
''φ'',
''θ'',
''ψ''. This presentation is convenient only for rotations about a fixed point.
*
Axis–angle representation (pictured at the right) specifies an angle with the axis about which the rotation takes place. It can be easily visualised. There are two variants to represent it:
** as a pair consisting of the angle and 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 v ...
for the axis, or
** as a
Euclidean vector
In mathematics, physics, and engineering, a Euclidean vector or simply a vector (sometimes called a geometric vector or spatial vector) is a geometric object that has magnitude (or length) and direction. Vectors can be added to other vectors ...
obtained by multiplying the angle with this unit vector, called the ''rotation vector'' (although, strictly speaking, it is a
pseudovector
In physics and mathematics, a pseudovector (or axial vector) is a quantity that is defined as a function of some vectors or other geometric shapes, that resembles a vector, and behaves like a vector in many situations, but is changed into its ...
).
* Matrices, versors (quaternions), and other
algebra
Algebra () is one of the broad areas of mathematics. Roughly speaking, algebra is the study of mathematical symbols and the rules for manipulating these symbols in formulas; it is a unifying thread of almost all of mathematics.
Elementary ...
ic things: see the section
''Linear and Multilinear Algebra Formalism'' for details.
A general rotation in
four dimensions has only one fixed point, the centre of rotation, and no axis of rotation; see
rotations in 4-dimensional Euclidean space for details. Instead the rotation has two mutually orthogonal planes of rotation, each of which is fixed in the sense that points in each plane stay within the planes. The rotation has two angles of rotation, one for each
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 rotation it can be used in other dimensions, such as ...
, through which points in the planes rotate. If these are and then all points not in the planes rotate through an angle between and . Rotations in four dimensions about a fixed point have six degrees of freedom. A four-dimensional direct motion in general position ''is'' a rotation about certain point (as in all
even
Even may refer to:
General
* Even (given name), a Norwegian male personal name
* Even (surname)
* Even (people), an ethnic group from Siberia and Russian Far East
**Even language, a language spoken by the Evens
* Odd and Even, a solitaire game wh ...
Euclidean dimensions), but screw operations exist also.
Linear and multilinear algebra formalism
When one considers motions of the Euclidean space that preserve
the origin, the
distinction between points and vectors, important in pure mathematics, can be erased because there is a canonical
one-to-one correspondence between points and
position vectors. The same is true for geometries other than
Euclidean, but whose space is an
affine space with a supplementary
structure
A structure is an arrangement and organization of interrelated elements in a material object or system, or the object or system so organized. Material structures include man-made objects such as buildings and machines and natural objects such a ...
; see
an example below. Alternatively, the vector description of rotations can be understood as a parametrization of geometric rotations
up to their composition with translations. In other words, one vector rotation presents many
equivalent rotations about ''all'' points in the space.
A motion that preserves the origin is the same as a
linear operator on vectors that preserves the same geometric structure but expressed in terms of vectors. For
Euclidean vector
In mathematics, physics, and engineering, a Euclidean vector or simply a vector (sometimes called a geometric vector or spatial vector) is a geometric object that has magnitude (or length) and direction. Vectors can be added to other vectors ...
s, this expression is their ''magnitude'' (
Euclidean norm). In
components
Circuit Component may refer to:
•Are devices that perform functions when they are connected in a circuit.
In engineering, science, and technology Generic systems
*System components, an entity with discrete structure, such as an assemb ...
, such operator is expressed with
orthogonal matrix that is multiplied to
column vectors.
As it
was already stated, a (proper) rotation is different from an arbitrary fixed-point motion in its preservation of the orientation of the vector space. Thus, the
determinant
In mathematics, the determinant is a scalar value that is a function of the entries of a square matrix. It characterizes some properties of the matrix and the linear map represented by the matrix. In particular, the determinant is nonzero if a ...
of a rotation orthogonal matrix must be 1. The only other possibility for the determinant of an orthogonal matrix is , and this result means the transformation is a
hyperplane reflection, a
point reflection (for
odd
Odd means unpaired, occasional, strange or unusual, or a person who is viewed as eccentric.
Odd may also refer to:
Acronym
* ODD (Text Encoding Initiative) ("One Document Does it all"), an abstracted literate-programming format for describing X ...
), or another kind of
improper rotation. Matrices of all proper rotations form the
special orthogonal group.
Two dimensions
In two dimensions, to carry out a rotation using a matrix, the point to be rotated counterclockwise is written as a column vector, then multiplied by a
rotation matrix In linear algebra, a rotation matrix is a transformation matrix that is used to perform a rotation in Euclidean space. For example, using the convention below, the matrix
:R = \begin
\cos \theta & -\sin \theta \\
\sin \theta & \cos \theta
\ ...
calculated from the angle :
:
.
The coordinates of the point after rotation are , and the formulae for and are
:
The vectors
and
have the same magnitude and are separated by an angle as expected.
Points on the plane can be also presented as
complex number
In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the fo ...
s: the point in the plane is represented by the complex number
:
This can be rotated through an angle by multiplying it by , then expanding the product using
Euler's formula
Euler's formula, named after Leonhard Euler, is a mathematical formula in complex analysis that establishes the fundamental relationship between the trigonometric functions and the complex exponential function. Euler's formula states that ...
as follows:
:
and equating real and imaginary parts gives the same result as a two-dimensional matrix:
:
Since complex numbers form a
commutative ring
In mathematics, a commutative ring is a ring in which the multiplication operation is commutative. The study of commutative rings is called commutative algebra. Complementarily, noncommutative algebra is the study of ring properties that are not ...
, vector rotations in two dimensions are commutative, unlike in higher dimensions. They have only one
degree of freedom, as such rotations are entirely determined by the angle of rotation.
Three dimensions
As in two dimensions, a matrix can be used to rotate a point to a point . The matrix used is a matrix,
:
This is multiplied by a vector representing the point to give the result
:
The set of all appropriate matrices together with the operation of
matrix multiplication is the
rotation group SO(3)
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 ...
. The matrix is a member of the three-dimensional
special orthogonal group, , that is it is an
orthogonal matrix with
determinant
In mathematics, the determinant is a scalar value that is a function of the entries of a square matrix. It characterizes some properties of the matrix and the linear map represented by the matrix. In particular, the determinant is nonzero if a ...
1. That it is an orthogonal matrix means that its rows are a set of orthogonal
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 v ...
s (so they are an
orthonormal basis
In mathematics, particularly linear algebra, an orthonormal basis for an inner product space ''V'' with finite dimension is a basis for V whose vectors are orthonormal, that is, they are all unit vectors and orthogonal to each other. For examp ...
) as are its columns, making it simple to spot and check if a matrix is a valid rotation matrix.
Above-mentioned Euler angles and axis–angle representations can be easily converted to a rotation matrix.
Another possibility to represent a rotation of three-dimensional Euclidean vectors are quaternions described below.
Quaternions
Unit
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 quater ...
s, or ''
versors'', are in some ways the least intuitive representation of three-dimensional rotations. They are not the three-dimensional instance of a general approach. They are more compact than matrices and easier to work with than all other methods, so are often preferred in real-world applications.
A versor (also called a ''rotation quaternion'') consists of four real numbers, constrained so the
norm of the quaternion is 1. This constraint limits the degrees of freedom of the quaternion to three, as required. Unlike matrices and complex numbers two multiplications are needed:
:
where is the versor, is its
inverse
Inverse or invert may refer to:
Science and mathematics
* Inverse (logic), a type of conditional sentence which is an immediate inference made from another conditional sentence
* Additive inverse (negation), the inverse of a number that, when a ...
, and is the vector treated as a quaternion with zero
scalar part. The quaternion can be related to the rotation vector form of the axis angle rotation by the
exponential map over the quaternions,
:
where is the rotation vector treated as a quaternion.
A single multiplication by a versor,
either left or right, is itself a rotation, but in four dimensions. Any four-dimensional rotation about the origin can be represented with two quaternion multiplications: one left and one right, by two ''different'' unit quaternions.
Further notes
More generally, coordinate rotations in any dimension are represented by orthogonal matrices. The set of all orthogonal matrices in dimensions which describe proper rotations (determinant = +1), together with the operation of matrix multiplication, forms the
special orthogonal group .
Matrices are often used for doing transformations, especially when a large number of points are being transformed, as they are a direct representation of the
linear operator. Rotations represented in other ways are often converted to matrices before being used. They can be extended to represent rotations and transformations at the same time using
homogeneous coordinates
In mathematics, homogeneous coordinates or projective coordinates, introduced by August Ferdinand Möbius in his 1827 work , are a system of coordinates used in projective geometry, just as Cartesian coordinates are used in Euclidean geometr ...
.
Projective transformations are represented by matrices. They are not rotation matrices, but a transformation that represents a Euclidean rotation has a rotation matrix in the upper left corner.
The main disadvantage of matrices is that they are more expensive to calculate and do calculations with. Also in calculations where
numerical instability
In the mathematical subfield of numerical analysis, numerical stability is a generally desirable property of numerical algorithms. The precise definition of stability depends on the context. One is numerical linear algebra and the other is algorit ...
is a concern matrices can be more prone to it, so calculations to restore
orthonormality, which are expensive to do for matrices, need to be done more often.
More alternatives to the matrix formalism
As was demonstrated above, there exist three
multilinear algebra rotation formalisms: one with
U(1), or complex numbers, for two dimensions, and two others with
versors, or quaternions, for three and four dimensions.
In general (even for vectors equipped with a non-Euclidean Minkowski
quadratic form
In mathematics, a quadratic form is a polynomial with terms all of degree two ("form" is another name for a homogeneous polynomial). For example,
:4x^2 + 2xy - 3y^2
is a quadratic form in the variables and . The coefficients usually belong to ...
) the rotation of a vector space can be expressed as a
bivector. This formalism is used in
geometric algebra and, more generally, in the
Clifford algebra representation of Lie groups.
In the case of a positive-definite Euclidean quadratic form, the double
covering group of the isometry group
is known as 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 ...
,
. It can be conveniently described in terms of a Clifford algebra. Unit quaternions give the group
.
In non-Euclidean geometries
In
spherical geometry, a direct motion of the
-sphere (an example of the
elliptic geometry) is the same as a rotation of -dimensional Euclidean space about the origin (). For odd , most of these motions do not have fixed points on the -sphere and, strictly speaking, are not rotations ''of the sphere''; such motions are sometimes referred to as ''
Clifford translations''. Rotations about a fixed point in elliptic and
hyperbolic geometries are not different from Euclidean ones.
Affine geometry and
projective geometry have not a distinct notion of rotation.
In relativity
A generalization of a rotation applies in
special relativity
In physics, the special theory of relativity, or special relativity for short, is a scientific theory regarding the relationship between space and time. In Albert Einstein's original treatment, the theory is based on two postulates:
# The law ...
, where it can be considered to operate on a four-dimensional space,
spacetime
In physics, spacetime is a mathematical model that combines the three dimensions of space and one dimension of time into a single four-dimensional manifold. Spacetime diagrams can be used to visualize relativistic effects, such as why differ ...
, spanned by three space dimensions and one of time. In special relativity, this space is called
Minkowski space, and the four-dimensional rotations, called
Lorentz transformations, have a physical interpretation. These transformations preserve a quadratic form called the
spacetime interval.
If a rotation of Minkowski space is in a space-like plane, then this rotation is the same as a spatial rotation in Euclidean space. By contrast, a rotation in a plane spanned by a space-like dimension and a time-like dimension is a
hyperbolic rotation, and if this plane contains the time axis of the reference frame, is called a "Lorentz boost". These transformations demonstrate the
pseudo-Euclidean nature of the Minkowski space. Hyperbolic rotations are sometimes described as ''
squeeze mappings'' and frequently appear on
Minkowski diagrams that visualize (1 + 1)-dimensional pseudo-Euclidean geometry on planar drawings. The study of relativity is deals with the
Lorentz group generated by the space rotations and hyperbolic rotations.
[Hestenes 1999, pp. 580–588.]
Whereas rotations, in physics and astronomy, correspond to rotations of
celestial sphere as a
2-sphere in the Euclidean 3-space, Lorentz transformations from induce
conformal transformations of the celestial sphere. It is a broader class of the sphere transformations known as
Möbius transformations.
Discrete rotations
Importance
Rotations define important classes of
symmetry
Symmetry (from grc, συμμετρία "agreement in dimensions, due proportion, arrangement") in everyday language refers to a sense of harmonious and beautiful proportion and balance. In mathematics, "symmetry" has a more precise definiti ...
:
rotational symmetry
Rotational symmetry, also known as radial symmetry in geometry, is the property a shape has when it looks the same after some rotation by a partial turn. An object's degree of rotational symmetry is the number of distinct orientations in which ...
is an
invariance with respect to a ''particular rotation''. The
circular symmetry is an invariance with respect to all rotation about the fixed axis.
As was stated above, Euclidean rotations are applied to
rigid body dynamics. Moreover, most of mathematical formalism in
physics
Physics is the natural science that studies matter, its fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge which ...
(such as the
vector calculus) is rotation-invariant; see
rotation for more physical aspects. Euclidean rotations and, more generally, Lorentz symmetry
described above are thought to be
symmetry laws of nature. In contrast, the
reflectional symmetry
In mathematics, reflection symmetry, line symmetry, mirror symmetry, or mirror-image symmetry is symmetry with respect to a reflection. That is, a figure which does not change upon undergoing a reflection has reflectional symmetry.
In 2D th ...
is not a precise symmetry law of nature.
Generalizations
The
complex-valued matrices analogous to real orthogonal matrices are the
unitary matrices , which represent rotations in complex space. The set of all unitary matrices in a given dimension forms a
unitary group of degree ; and its subgroup representing proper rotations (those that preserve the orientation of space) is the
special unitary group of degree . These complex rotations are important in the context of
spinors. The elements of
are used to parametrize ''three''-dimensional Euclidean rotations (see
above), as well as respective transformations of the
spin (see
representation theory of SU(2)).
See also
*
Aircraft principal axes
*
Charts on SO(3)
*
Coordinate rotations and reflections
*
CORDIC algorithm
*
Hyperbolic rotation
*
Infinitesimal rotation
*
Irrational rotation
*
Orientation (geometry)
*
Rodrigues' rotation formula
*
Rotation of axes
*
Vortex
In fluid dynamics, a vortex ( : vortices or vortexes) is a region in a fluid in which the flow revolves around an axis line, which may be straight or curved. Vortices form in stirred fluids, and may be observed in smoke rings, whirlpools in ...
Footnotes
References
*
*
*
{{Computer graphics
Euclidean symmetries
Rotational symmetry
Linear operators
Unitary operators