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Four-dimensional number system
In mathematics, the quaternions form a number system similar to the complex numbers, with the usual arithmetical operations of addition, subtraction,
Quaternion
Unofficial coat of arms of the Holy Roman Empire
The Quaternion Eagle (German: Quaternionenadler; Italian: aquila quaternione), also known as the Imperial Quaternion Eagle (German: Quaternionen-Reichsadler)
Quaternion_Eagle
Topics referred to by the same term
up quaternion in Wiktionary, the free dictionary. Quaternion can refer to: Quaternions, a number system that extends the complex numbers. Quaternion rotation
Quaternion_(disambiguation)
Correspondence between quaternions and 3D rotations
Unit quaternions, known as versors, provide a convenient mathematical notation for representing spatial orientations and rotations of elements in three
Quaternions and spatial rotation
Quaternions_and_spatial_rotation
Non-abelian group of order eight
In group theory, the quaternion group Q8 (sometimes just denoted by Q) is a non-abelian group of order eight, isomorphic to the eight-element subset {
Quaternion_group
Generalization of Gaussian integers to quaternions
In mathematics, a Hurwitz quaternion (or Hurwitz integer) is a quaternion whose components are either all integers or all half-integers (halves of odd
Hurwitz_quaternion
Quaternion is a poetry style in which the theme is divided into four parts. Each part of a quaternion explores the complementary natures of the theme
Quaternion_(poetry)
Generalization of quaternions to other fields
In mathematics, a quaternion algebra over a field F is a central simple algebra A over F that has dimension 4 over F. Every quaternion algebra becomes a
Quaternion_algebra
Four-dimensional associative algebra over the reals
In abstract algebra, the split-quaternions or coquaternions form an algebraic structure introduced by James Cockle in 1849 under the latter name. They
Split-quaternion
Eight-dimensional algebra over the real numbers
In mathematics, the dual quaternions are an 8-dimensional real algebra isomorphic to the tensor product of the quaternions and the dual numbers. Thus
Dual_quaternion
In mathematics, quaternions are a non-commutative number system that extends the complex numbers. Quaternions and their applications to rotations were
History_of_quaternions
Mathematical strategy
angles and unit quaternions. This article explains how to convert between the two representations. Actually this simple use of "quaternions" was first presented
Conversion between quaternions and Euler angles
Conversion_between_quaternions_and_Euler_angles
In differential geometry, a quaternion-Kähler manifold (or quaternionic Kähler manifold) is a Riemannian 4n-manifold whose Riemannian holonomy group is
Quaternion-Kähler_manifold
Quaternion of norm 1 (unit quaternion)
In mathematics, a versor is a quaternion whose norm is one, also known as a unit quaternion. Each versor has the form u = exp ( a r ) = cos a +
Versor
Mutation of quaternions where unit vectors square to +1
In abstract algebra, the algebra of hyperbolic quaternions is a nonassociative algebra over the real numbers with elements of the form q = a + b i + c
Hyperbolic_quaternion
Hamilton's original treatment of quaternions
Hamilton invented quaternions, a mathematical entity, in 1843. This article describes Hamilton's original treatment of quaternions, using his notation
Classical Hamiltonian quaternions
Classical_Hamiltonian_quaternions
Function used in computer graphics
Ken Shoemake for animating three-dimensional rotations, represented as quaternions on an abstract 3-sphere. When the interpolation parameter represents
Spherical linear interpolation
Spherical_linear_interpolation
Ways to represent 3D rotations
representing a rotation as numbers in a computer, some people prefer the quaternion representation or the axis+angle representation, because they avoid the
Rotation formulations in three dimensions
Rotation_formulations_in_three_dimensions
Constituent state of the Holy Roman Empire with representation in the Imperial Diet
Bench of the Rhine. The so-called imperial quaternions (German: Quaternionen der Reichsverfassung "quaternions of the imperial constitution"; from Latin
Imperial_Estate
Function theory with quaternion variable
the study of functions with quaternions as the domain and/or range. Such functions can be called functions of a quaternion variable just as functions of
Quaternionic_analysis
Senior society at Furman University, US
Quaternion Senior Order (QSO) is an honor society for seniors at Furman University in Greenville, South Carolina, US. It was formed in 2023 by the merger
Quaternion_Senior_Order
Motion of a certain space that preserves at least one point
(although, strictly speaking, it is a pseudovector). Matrices, versors (quaternions), and other algebraic things: see the section Linear and Multilinear
Rotation_(mathematics)
rendition of the coat of the empire was the "Quaternion Eagle" (so named after the imperial quaternions) printed by David de Negker of Augsburg, after
Coats of arms of the Holy Roman Empire
Coats_of_arms_of_the_Holy_Roman_Empire
Position of something in relation to its surroundings
axis–angle representation. Other widely used methods include rotation quaternions, rotors, Euler angles, or rotation matrices. More specialist uses include
Orientation_(geometry)
Irish mathematician and physicist (1805–1865)
research included the analysis of geometrical optics, Fourier analysis, and quaternions, the last of which made him one of the founders of modern linear algebra
William_Rowan_Hamilton
Matrix representing a Euclidean rotation
unit quaternions. Multiplication of rotation matrices is homomorphic to multiplication of quaternions, and multiplication by a unit quaternion rotates
Rotation_matrix
Differential geometry concept
In differential geometry, a quaternion-Kähler symmetric space or Wolf space is a quaternion-Kähler manifold which, as a Riemannian manifold, is a Riemannian
Quaternion-Kähler symmetric space
Quaternion-Kähler_symmetric_space
Concept in mathematics
complex projective space, to the case where coordinates lie in the ring of quaternions H . {\displaystyle \mathbb {H} .} Quaternionic projective space of dimension
Quaternionic_projective_space
Algorithm to solve Wahba's problem
The quaternion estimator algorithm (QUEST) is an algorithm designed to solve Wahba's problem, that consists of finding a rotation matrix between two coordinate
Quaternion estimator algorithm
Quaternion_estimator_algorithm
Quaternions with complex number coefficients
variants thereof, and the elements of {1, i, j, k} multiply as in the quaternion group and commute with their coefficients. There are three types of biquaternions
Biquaternion
Group of unitary complex matrices with determinant of 1
is isomorphic to the group of quaternions of norm 1, and is thus diffeomorphic to the 3-sphere. Since unit quaternions can be used to represent rotations
Special_unitary_group
Method for producing composition algebras
process are known as Cayley–Dickson algebras, for example complex numbers, quaternions, and octonions. These examples are useful composition algebras frequently
Cayley–Dickson_construction
French politician and mathematician
published two works in geometric algebra, Introduction à la Méthode des Quaternions (1881) and Théorie et applications des equipollences (1887). He also
Charles-Ange_Laisant
3D model's surface projected to a 2D image
in model space, while "W" (in addition to XYZ) is used in calculating quaternion rotations, a common operation in computer graphics. UV texturing permits
UV_mapping
Comprehensive physical model
left and right-handed 4 × 4 quaternion matrices is equivalent to including a single right-multiplication by a unit quaternion which adds an extra SU(2)
Grand_Unified_Theory
Every natural number can be represented as the sum of four integer squares
Hurwitz quaternions, which are the analog of integers for quaternions. Proof using the Hurwitz integers The Hurwitz quaternions consist of all quaternions with
Lagrange's four-square theorem
Lagrange's_four-square_theorem
Four-dimensional algebra over the real numbers
The planar quaternions make up a four-dimensional algebra over the real numbers. Their primary application is in representing rigid body motions in 2D
Applications of dual quaternions to 2D geometry
Applications_of_dual_quaternions_to_2D_geometry
Eighth letter of the Latin alphabet
constant H {\displaystyle \mathbb {H} } : Blackboard bold capital H used in quaternion notation 1 Also for encodings based on ASCII, including the DOS, Windows
H
Nonabelian group in algebraic group theory
subgroup of the unit quaternions, under the isomorphism Spin(3) ≅ Sp(1), where Sp(1) is the multiplicative group of unit quaternions. (For a description
Binary_tetrahedral_group
Special interest group of mathematicians (1899 to 1913)
The Quaternion Association was a scientific society, self-described as an "International Association for Promoting the Study of Quaternions and Allied
Quaternion_Association
Linear transformation of spacetime coordinates
complex quaternions or the complexified quaternions or even just the quaternions in the literature. The biquaternions differ from the quaternions only in
Biquaternion Lorentz transformation
Biquaternion_Lorentz_transformation
Functions of complex quaternions
Functions in the complex plane can be extended to functions of complex quaternions (biquaternions). This is simple when the function can be expressed as
Biquaternion_functions
Element of a unital algebra over the field of real numbers
representation theory. In the nineteenth century, number systems called quaternions, tessarines, coquaternions, biquaternions, and octonions became established
Hypercomplex_number
Branch of mathematics that studies the properties of groups
Cyclic group Zn Symmetric group Sn Alternating group An Dihedral group Dn Quaternion group Q Cauchy's theorem Lagrange's theorem Sylow theorems Hall's theorem
Group_theory
Algebra based on a vector space with a quadratic form
subspace. As K-algebras, they generalize the real numbers, complex numbers, quaternions and several other hypercomplex number systems. The theory of Clifford
Clifford_algebra
groups are a special class of Kleinian groups constructed using orders in quaternion algebras. They are particular instances of arithmetic groups. An arithmetic
Arithmetic hyperbolic 3-manifold
Arithmetic_hyperbolic_3-manifold
Group of printed leaves, folded in the middle and bound together into a book binding
folded sheets of vellum or parchment, i.e. 8 leaves, 16 sides. The term quaternion (or sometimes quaternum) designates such a unit. A gathering made of a
Section_(bookbinding)
Measure of distance between atoms of superimposed proteins
minimize the RMSD. Coutsias, et al. presented a simple derivation, based on quaternions, for the optimal solid body transformation (rotation-translation) that
Root mean square deviation of atomic positions
Root_mean_square_deviation_of_atomic_positions
Bridge in Dublin
discovery of the quaternions, Hamilton's sons William Edwin Hamilton and Archibald Henry Hamilton referred to the bridge as the Quaternion Bridge. Given
Broom_Bridge
Mathematical group
2 {\displaystyle x^{3}-2} over Q . {\displaystyle \mathbb {Q} .} The Quaternion group can be found as the Galois group of a field extension of Q {\displaystyle
Galois_group
Non-Euclidean geometry
arcs. The first success of quaternions was a rendering of spherical trigonometry to algebra. Hamilton called a quaternion of norm one a versor, and these
Elliptic_geometry
Event commemorating the Irish mathematician Hamilton's discovery of quaternions
Rowan Hamilton discovered the non-commutative algebraic system known as quaternions, while walking with his wife along the banks of the Royal Canal. The
Hamilton_Walk
Mathematical object
:\|q\|=1\right\}.} This description as the quaternions of norm one identifies the 3-sphere with the versors in the quaternion division ring. Just as the unit circle
3-sphere
Anglo-American poet (1612–1672)
(quaternion) Four Seasons of the Year (quaternion) Four Elements (quaternion) Of The Four Ages of Man (quaternion) The Four Monarchies (quaternion) The
Anne_Bradstreet
Type of matrix representation
[0, ∞). The polar decomposition of quaternions H {\displaystyle \mathbb {H} } with orthonormal basis quaternions 1 , ı ^ , ȷ ^ , k ^ {\displaystyle 1
Polar_decomposition
Mathematical theorem in the study of analysis
C(X, H) of quaternion-valued continuous functions on the compact space X, again with the topology of uniform convergence. If a quaternion q is written
Stone–Weierstrass_theorem
Movement with a fixed point is rotation
four numbers is called a quaternion. While the quaternion described above does not involve complex numbers, if quaternions are used to describe two successive
Euler's_rotation_theorem
Mathematical operation
impedance matching of transmission lines. In the four-dimensional space of quaternions a + b i → + c j → + d k → {\displaystyle a+b{\vec {i}}+c{\vec {j}}+d{\vec
Cayley_transform
Type of algorithm
Alternatively, optimal rotation matrix can also be directly evaluated as quaternion. This alternative description has been used in the development of a rigorous
Kabsch_algorithm
Theorem: (cos x + i sin x)^n = cos nx + i sin nx
\end{aligned}}} To find the roots of a quaternion there is an analogous form of de Moivre's formula. A quaternion in the form q = d + a i ^ + b j ^ + c
De_Moivre's_formula
Heraldic eagle used in Germany and Austria
in the Wernigerode Armorial (c. 1490) Quaternion Eagle c. 1510, the eagle displaying the imperial quaternions on its remiges. The imperial eagle depicted
Reichsadler
French mathematician (1823–1886)
associativity, distribution, and inverses) and used the algebra of quaternions and versors to describe spherical trigonometry. However, in 1890 P. G
Jules_Hoüel
Connected non-abelian Lie group lacking nontrivial connected normal subgroups
map homomorphism from SU(2) × SU(2) to SO(4) given by quaternion multiplication; see quaternions and spatial rotation. Thus SO(4) is not a simple group
Simple_Lie_group
In mathematics, a Riemann surface
{\displaystyle 2} quotient of the group of norm- 1 {\displaystyle 1} elements of a quaternion algebra, but the ( 3 , 3 , 4 ) {\displaystyle (3,3,4)} group does. Under
Bolza_surface
German mathematician (1832–1903)
Mechanik (Berlin, 1876). Cauchy–Lipschitz theorem Lipschitz domain Lipschitz quaternion Lipschitz continuity Uniform, Hölder and Lipschitz continuity Lipschitz
Rudolf_Lipschitz
Loss of one degree of freedom in a three-dimensional, three-gimbal mechanism
system) and integrating sensed rotation and acceleration digitally using quaternion methods to derive vehicle orientation and velocity. Another way to replace
Gimbal_lock
Special orthogonal group
respectively by left- and right-multiplication by unit quaternions; see the paragraph "Relation to quaternions" below. The four rotations are pairwise different
Rotations in 4-dimensional Euclidean space
Rotations_in_4-dimensional_Euclidean_space
Group in which the order of every element is a power of p
very dissimilar from the quaternion groups and the semidihedral groups. Together the dihedral, semidihedral, and quaternion groups form the 2-groups of
P-group
Elements of a field, e.g. real numbers, in the context of linear algebra
a 1 × 1 matrix, is often said to be a scalar. The real component of a quaternion is also called its scalar part. The term scalar matrix is used to denote
Scalar_(mathematics)
Mathematical concept
abelian group is nilpotent. For a small non-abelian example, consider the quaternion group Q8, which is a smallest non-abelian p-group. It has center {1, −1}
Nilpotent_group
Nonabelian group of order 120
algebra of quaternions, the binary icosahedral group is concretely realized as a discrete subgroup of the versors, which are the quaternions of norm one
Binary_icosahedral_group
Transformations induced by a mathematical group
quaternions and spatial rotation. This is not a faithful action because the quaternion −1 leaves all points where they were, as does the quaternion 1
Group_action
Hypercomplex number system
Octonions have eight dimensions; twice the number of dimensions of the quaternions, of which they are an extension. They are noncommutative and nonassociative
Octonion
Group of rotations in 3 dimensions
be understood as the group of versors (quaternions with absolute value 1). The connection between quaternions and rotations, commonly exploited in computer
3D_rotation_group
Sum of directed areas in exterior algebra
complex numbers in two dimensions and to both pseudovectors and vector quaternions in three dimensions. They can be used to generate rotations in a space
Bivector
Monster and modular connection
Cyclic group Zn Symmetric group Sn Alternating group An Dihedral group Dn Quaternion group Q Cauchy's theorem Lagrange's theorem Sylow theorems Hall's theorem
Monstrous_moonshine
Index of articles associated with the same name
Cyclic group Zn Symmetric group Sn Alternating group An Dihedral group Dn Quaternion group Q Cauchy's theorem Lagrange's theorem Sylow theorems Hall's theorem
Janko_group
Concept in mathematics
The Hurwitz quaternion order is a specific order in a quaternion algebra over a suitable number field. The order is of particular importance in Riemann
Hurwitz_quaternion_order
Specific set of Hamiltonian quaternions with the same symmetry as the 600-cell
In mathematics, the icosians are a specific set of Hamiltonian quaternions with the same symmetry as the 600-cell. The term can be used to refer to two
Icosian
Function that is its own inverse
C*-algebras are special types of Banach algebras with involutions. In a quaternion algebra, an (anti-)involution is defined by the following axioms: if we
Involution_(mathematics)
Mathematical equation linking e, i and π
where n = 2. A similar identity also applies to quaternion exponential: let {i, j, k} be the basis quaternions; then, e 1 3 ( i ± j ± k ) π + 1 = 0. {\displaystyle
Euler's_identity
Holy Roman Empire drinking vessel
glass was decorated with a double-headed eagle, usually in the shape of a Quaternion Eagle. The Reichsadler means "Imperial Eagle" or double-headed eagle which
Imperial_Eagle_beaker
Mathematical group
quaternions of norm 1 in a suitable order in a quaternion algebra. More specifically, the triangle group is the quotient of the group of quaternions by
(2,3,7)_triangle_group
Island in the North Atlantic Ocean
Hamilton, famous for work in classical mechanics and the invention of quaternions. Francis Ysidro Edgeworth's contribution, the Edgeworth Box. remains
Ireland
Commutative group (mathematics)
Cyclic group Zn Symmetric group Sn Alternating group An Dihedral group Dn Quaternion group Q Cauchy's theorem Lagrange's theorem Sylow theorems Hall's theorem
Abelian_group
Mathematical descriptions of a rotation group
two unit quaternions of opposite sign, and, as in the space of rotations in three dimensions, the quaternion product of two unit quaternions will yield
Charts_on_SO(3)
Compact Riemann surface of genus 3
One chooses a suitable Hurwitz quaternion order Q H u r {\displaystyle {\mathcal {Q}}_{\mathrm {Hur} }} in the quaternion algebra, Γ(I) is then the group
Klein_quartic
Use of coordinates for representing vectors
around 1843, as he revealed quaternions, a system which uses vectors and scalars to span a four-dimensional space. For a quaternion q = a + bi + cj + dk, Hamilton
Vector_notation
Sporadic simple group
Cyclic group Zn Symmetric group Sn Alternating group An Dihedral group Dn Quaternion group Q Cauchy's theorem Lagrange's theorem Sylow theorems Hall's theorem
Janko_group_J3
Mathematical abelian group
B , A ∘ B } {\displaystyle \{e,A,B,A\circ B\}} forms a Klein group. Quaternion group List of small groups Vorlesungen über das Ikosaeder und die Auflösung
Klein_four-group
the multiplicative group of unit quaternions. (For a description of this homomorphism see the article on quaternions and spatial rotations.) Explicitly
Binary_octahedral_group
Type of group in group theory
by taking the unit groups of orders in quaternion algebras over number fields (for example the Hurwitz quaternion order). Similar constructions can be performed
Arithmetic_group
(pseudo-)Riemannian manifold whose geodesics are reversible
End(TM) isomorphic to the imaginary quaternions at each point, and compatible with the Riemannian metric, is called quaternion-Kähler symmetric space. An irreducible
Symmetric_space
Mathematical group that can be generated as the set of powers of a single element
Cyclic group Zn Symmetric group Sn Alternating group An Dihedral group Dn Quaternion group Q Cauchy's theorem Lagrange's theorem Sylow theorems Hall's theorem
Cyclic_group
Mathematical operation on vectors in 3D space
algebra of quaternions and the non-commutative Hamilton product. In particular, when the Hamilton product of two vectors (that is, pure quaternions with zero
Cross_product
Applied mathematics problem
_{k}} defines a circle of admissible quaternions, called a ''quaternion circle''. To see this, choose any unit quaternion q k {\displaystyle \mathbf {q} _{k}}
Wahba's_problem
Topics referred to by the same term
of the nineteenth century, including Quaternions Tessarines Coquaternions Biquaternions Hyperbolic quaternions This disambiguation page lists articles
Vector_algebra
Concept in mathematics
determinant defined for Hermitian matrices over a quaternion algebra, introduced by Moore (1922). Because quaternion multiplication does not commute, it is necessary
Moore determinant of a Hermitian matrix
Moore_determinant_of_a_Hermitian_matrix
Mathematical relation in spherical triangles
using quaternions. Let u, v, and w denote the unit vectors from the center of the unit sphere to those corners of the triangle. We define the quaternion u
Spherical_law_of_cosines
Tenth letter of the Latin alphabet
for Japan. In mathematics, j is one of the three imaginary units of quaternions. Also in mathematics, j is one of the three unit vectors. In the Metric
J
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