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Representation of a three-dimensional rotation
The conformal rotation vector, whose coordinates are also known as modified Rodrigues parameters or Wiener–Milenkovic parameters, is a three-dimensional
Conformal_rotation_vector
Ways to represent 3D rotations
the rotation angles. The stereographic projection of a unit quaternion onto the pure-imaginary hyperplane is called the conformal rotation vector, with
Rotation formulations in three dimensions
Rotation_formulations_in_three_dimensions
Vector field on a pseudo-Riemannian manifold that preserves the metric tensor
{\displaystyle \lambda } . The derivatives of one parameter families of conformal maps are conformal Killing fields. Killing tensor fields are symmetric tensor fields
Killing_vector_field
transforming vectors by matrix multiplication. The Lie group of these transformations has been called the conformal orthogonal group, the conformal linear transformation
Conformal linear transformation
Conformal_linear_transformation
Type of geometric algebra
so that the effect of a translation (or any conformal mapping) of the base space corresponds to a rotation in the higher-dimensional space. In the algebra
Conformal_geometric_algebra
Theorem limiting types of conformal mappings in Euclidean space of dimension > 2
in 1850, is a rigidity theorem about conformal mappings in Euclidean space. It states that every smooth conformal mapping on a domain of Rn, where n >
Liouville's theorem (conformal mappings)
Liouville's_theorem_(conformal_mappings)
Quantum field theory enjoying conformal symmetry
A conformal field theory (CFT) is a quantum field theory that is invariant under conformal transformations. In two dimensions, there is an infinite-dimensional
Conformal_field_theory
Motion of a certain space that preserves at least one point
and a unit vector for the axis, or as a Euclidean vector obtained by multiplying the angle with this unit vector, called the rotation vector (although
Rotation_(mathematics)
Tensor in general relativity
the Carter constant. Conformal Killing tensors are a generalization of Killing tensors and conformal Killing vectors. A conformal Killing tensor is a tensor
Killing_tensor
Special orthogonal group
plane for which every vector in the plane is unchanged after the rotation. An "invariant plane" is a plane for which every vector in the plane, although
Rotations in 4-dimensional Euclidean space
Rotations_in_4-dimensional_Euclidean_space
Physical quantity that changes sign with improper rotation
pseudovector (or axial vector) is a quantity that transforms like a vector under continuous rigid transformations such as rotations or translations, but
Pseudovector
Gravity theories that are invariant under Weyl transformations
Conformal gravity refers to gravity theories that are invariant under conformal transformations in the Riemannian geometry sense; more accurately, they
Conformal_gravity
Algebraic structure designed for geometry
the 4D null cone of the 5D CGA vector subspace. This allows all conformal transformations to be performed as rotations and reflections and is covariant
Geometric_algebra
Mathematical function, in linear algebra
simple examples include rotation and reflection linear transformations. Let V {\displaystyle V} and W {\displaystyle W} be vector spaces over the same field
Linear_map
conformal field theory called the critical O(n) model. This CFT can be analyzed using expansions in the dimension d or in n, or using the conformal bootstrap
N-vector_model
Type of group in mathematics
real orthogonal transforms preserve angles, and are thus conformal maps, though not all conformal linear transforms are orthogonal. In classical terms this
Orthogonal_group
Diagram of different points in spacetime
readable introduction to the concept of conformal infinity plus examples. Frauendiener, Jörg (2004). "Conformal Infinity". Living Reviews in Relativity
Penrose_diagram
Four-dimensional number system
the algebra, not just vectors and other quaternions, but also lines, planes, circles, spheres, rays, and so on. In the conformal model of Euclidean geometry
Quaternion
Application of Clifford algebra
the full projective group; this is unlike 3D Conformal Geometric Algebra, which contains the full conformal group. To a first approximation, the physical
Plane-based_geometric_algebra
Extension to the Poincaré group
accounts for rotations, translations, and boosts—into the more comprehensive conformal group. Conformal symmetry encompasses special conformal transformations
Conformal_symmetry
Branch of mathematics studying functions of a complex variable
orientation. Conformal maps preserve both angles and the shapes of infinitesimally small figures, but not necessarily their size or curvature. The conformal property
Complex_analysis
Family of linear transformations
transforms as the time component of a four-vector. It is a rotational scalar. The current density is a 3-vector. The Maxwell equations are invariant under
Lorentz_transformation
Concept in mathematical group theory
the conformal geometry of the space. Several specific conformal groups are particularly important: The conformal orthogonal group. If V is a vector space
Conformal_group
Object in geometric algebra
called Clifford algebra) of a vector space that represents a rotation about the origin. More precisely, for each rotation there exist two rotors that represent
Rotor_(mathematics)
Feature of a system that is preserved under some transformation
is an antisymmetric matrix (giving the Lorentz and rotational symmetries) and P is a general vector (giving the translational symmetries). Other symmetries
Symmetry_(physics)
Mathematical description of spacetime used in relativity
M-theory are two examples where n > 4. In string theory there appear conformal field theories with 1 + 1 spacetime dimensions. de Sitter space can be
Minkowski_spacetime
Characteristic property of holomorphic functions
function to be conformal. Moreover, because the composition of a conformal transformation with another conformal transformation is also conformal, the composition
Cauchy–Riemann_equations
Mathematical descriptions of a rotation group
of composition. By definition, a rotation about the origin is a linear transformation that preserves length of vectors (it is an isometry) and preserves
Charts_on_SO(3)
Geographic coordinate specifying north-south position
the Albers equal-area conic projection. The conformal latitude, χ, gives an angle-preserving (conformal) transformation to the sphere. χ ( ϕ ) = 2 tan
Latitude
Lie group of Lorentz transformations
represents conformal geometry on the sphere S2. The (identity component of the) Euclidean group SE(2) is the stabilizer of a null vector, so the homogeneous
Lorentz_group
Calculus of vector-valued functions
Vector calculus or vector analysis is a branch of mathematics concerned with the differentiation and integration of vector fields, primarily in three-dimensional
Vector_calculus
Matrices important in quantum mechanics and the study of spin
straightforward to likewise work out the adjoint action on the Pauli vector, namely rotation of any angle a {\displaystyle a} along any axis n ^ {\displaystyle
Pauli_matrices
Type of transformations applicable to coordinate space-time
include all conformal, one-to-one transformations on coordinate space-time. They are less studied in physics because, unlike the rotations and translations
Inversion_transformation
Bijection of a set using properties of shapes in space
Similarity Affine transformation Projective transformation Inversion Conformal transformations preserve angles, and are, in the first order, similarities
Geometric_transformation
Distance-preserving mathematical transformation
necessarily preserves angles, therefore a linear isometry transformation is a conformal linear transformation. Examples A linear map from C n {\displaystyle \mathbb
Isometry
Group that is also a differentiable manifold with group operations that are smooth
R 3 {\displaystyle \mathbb {R} ^{3}} , conformal geometry corresponds to enlarging the group to the conformal group, whereas in projective geometry one
Lie_group
Rational function of the form (az + b)/(cz + d)
bijective conformal orientation-preserving maps from the n-sphere to the n-sphere. Such a transformation is the most general form of conformal mapping of
Möbius_transformation
Transformation method within a three-dimensional space
reference system A by the following formula (position vector transformation convention and very small rotation angles simplification): [ X Y Z ] B = [ c x c y
Helmert_transformation
Modified theory of gravity developed by John Moffat
by a repulsive fifth force due to the vector field. STVG has been used successfully to explain galaxy rotation curves, the mass profiles of galaxy clusters
Scalar–tensor–vector_gravity
System for describing optical polarization
invented by R. C. Jones in 1941. Polarized light is represented by a Jones vector, and linear optical elements are represented by Jones matrices. When light
Jones_calculus
Set of coordinates where the coordinate hypersurfaces all meet at right angles
for generating orthogonal coordinates systems in two dimensions is by a conformal mapping of a standard two-dimensional grid of Cartesian coordinates (x
Orthogonal_coordinates
3D model's surface projected to a 2D image
{\arcsin(d_{y})}{\pi }}.} Cartographic projection Geodesic Least squares conformal map Mesh parameterization NURBS Polygon mesh Radon transformation Lightmap
UV_mapping
Hypothesis proposing a modification of Newton's laws
modified gravity version. Its primary motivation is to explain galaxy rotation curves without invoking dark matter, and is one of the most well-known
Modified_Newtonian_dynamics
Four-dimensional algebra over the real numbers
Affine transformation Projective plane Homogeneous coordinates SLERP Conformal geometric algebra Matsuda, Genki; Kaji, Shizuo; Ochiai, Hiroyuki (2014)
Applications of dual quaternions to 2D geometry
Applications_of_dual_quaternions_to_2D_geometry
Fiber bundle of the 3-sphere over the 2-sphere, with 1-spheres as fibers
fixing the tip of the z {\displaystyle z} vector does not specify the rotation fully; a further rotation is possible about the z {\displaystyle z} axis
Hopf_fibration
Branch of mathematics that studies the properties of groups
isometry group of X. If instead angles are preserved, one speaks of conformal maps. Conformal maps give rise to Kleinian groups, for example. Symmetries are
Group_theory
Overview of and topical guide to geometry
geometry Birational geometry Complex geometry Computational geometry Conformal geometry Constructive solid geometry Contact geometry Convex geometry
Outline_of_geometry
Type of symmetry in physics
preserve geodesics without necessarily preserving the affine parameter. A conformal vector field is one which satisfies: L X g = ϕ g {\displaystyle {\mathcal
Spacetime_symmetries
Doughnut-shaped surface of revolution
torus (total angle 2π/3). These are the only conformal equivalence classes of flat tori that have any conformal automorphisms other than those generated by
Torus
To rotate a QGA point, it must be projected to a vector or converted to a CGA point for rotation operations, then the rotated result can be re-embedded
Quadric_geometric_algebra
Analogies between Maxwell's and Einstein's field equations
generation due to rotation. Fluid mechanics – rotational fluid drag of a solid sphere immersed in fluid, analogous directions and senses of rotation as magnetism
Gravitoelectromagnetism
State space for internal degrees of freedom of a subatomic particle
other groups might be considered: Conformal symmetry: For pseudo-Euclidean space, symmetries are described by the conformal group Conf ( p , q ) ≅ O ( p
Multiplet
Proposed theories of gravity
doi:10.1002/mana.19540120302. ISSN 0025-584X. Littlewood, D. E. (1953). "Conformal transformations and kinematical relativity". Mathematical Proceedings
Alternatives to general relativity
Alternatives_to_general_relativity
Topic in general relativity
Kronecker delta. We can instead make use of conformal time as the time component yielding the longitudinal or conformal Newtonian gauge: d s 2 = a 2 ( τ ) [
Newtonian_gauge
Physics property associated with symmetries
supersymmetry. In conformal field theory: The central charge of the Virasoro algebra, sometimes referred to as the conformal central charge or the conformal anomaly
Charge_(physics)
Physical quantities taking values at each point in space and time
this vector transform between themselves contravariantly under rotations in space. Similarly, a dual (or co-) vector field attaches a dual vector to each
Field_(physics)
Algebraic structure used in analysis
Lie group of rotations of space, and each vector v ∈ R 3 {\displaystyle v\in \mathbb {R} ^{3}} may be pictured as an infinitesimal rotation around the axis
Lie_algebra
Setting of relativistic physics in geometric algebra
algebra is a vector space that allows not only vectors, but also bivectors (directed quantities describing rotations associated with rotations or particular
Spacetime_algebra
Exterior algebraic map taking tensors from p forms to n-p forms
the axis, with speed equal to the length of the axis of rotation. A scalar product on a vector space V {\displaystyle V} gives an isomorphism V ≅ V ∗ {\displaystyle
Hodge_star_operator
Science of measuring the shape, orientation, and gravity of Earth
surface without deformation. The compromise most often chosen — called a conformal projection — preserves angles and length ratios so that small circles
Geodesy
Pair of counter-rotating wedge prisms used for optical beam steering
rotated independently about the optical axis. By varying the relative rotation angles of the wedges, the device deflects an incident beam to any azimuth
Risley_prisms
Generalization of perpendicularity
that transforming orthogonal vectors by the same conformal linear transformation will keep those vectors orthogonal. Two vector subspaces A {\displaystyle
Orthogonality_(mathematics)
Approach to quantum gravity utilizing Wick rotations
rotating the vector representing that number by an angle of π / 2 {\displaystyle \pi /2} radians about the origin. For example, a Wick rotation could be used
Euclidean_quantum_gravity
Concept in physics
theory of gravitation that tries to explain the observation of the flat rotation curves of galaxies. In general relativity, the gravitational field is characterized
Nonsymmetric gravitational theory
Nonsymmetric_gravitational_theory
64-bit extension of the ARM architecture
Doubling Multiply Subtract, Returning High Half. The instructions are added in vector and scalar forms. A set of AArch64 load and store instructions that can
AArch64
Mathematics of smooth surfaces
differential equations to prove existence. In fact the Ricci flow on conformal metrics on S2 is defined on functions u(x, t) by u t = 4 π − K ′ ( x
Differential geometry of surfaces
Differential_geometry_of_surfaces
Nonlinear differential operator used to study conformal mappings
derivative as a measure of how much a conformal map deviates from a Möbius transformation. Let f {\displaystyle f} be a conformal mapping in a neighborhood of
Schwarzian_derivative
Machine learning technique useful for dimensionality reduction
has the ability to control the growth of the GSOM. The conformal map approach uses conformal mapping to interpolate each training sample between grid
Self-organizing_map
Model of n-dimensional hyperbolic geometry
hyperboloid in (n+1)-dimensional Minkowski space or by the displacement vectors from the origin to those points, and m-planes are represented by the intersections
Hyperboloid_model
Relativistic generalization of Mordehai Milgrom's MOND paradigm
Tensor–vector–scalar gravity (TeVeS), developed by Jacob Bekenstein in 2004, is a relativistic generalization of Mordehai Milgrom's Modified Newtonian
Tensor–vector–scalar_gravity
Attraction of masses and energy
of the body and is modified by the centrifugal effects arising from the rotation of the body. In this context, gravity gives weight to physical objects
Gravity
Phenomenon in particle physics
partners of the nucleon appear. Chiral symmetry breaking and the quantum conformal anomaly account for approximately 99% of the mass of a proton or neutron
Chiral_symmetry_breaking
Class of partial differential equations
its Applications 59 (2018): 1-11. Flory, Mario, and Michal P. Heller. "Conformal field theory complexity from Euler-Arnold equations." Journal of High
Euler–Arnold_equation
Classical statement of gravity as force
Press 1999 ISBN 0-520-08816-6 ISBN 0-520-08817-4 "Rotational Flattening". farside.ph.utexas.edu. The vector difference r2 − r1 points from object 1 to object
Newton's law of universal gravitation
Newton's_law_of_universal_gravitation
Concept in group theory (mathematics)
is also valid in their subgroups, e.g. orthogonal, pseudo-Euclidean, conformal, and classical groups. Because the elements of Pin groups are the composition
Invariant_decomposition
Physical theory with fields invariant under the action of local "gauge" Lie groups
significance, such as a velocity or an axis of rotation, its representation as numbers arranged in a vector or matrix is also changed by a coordinate transformation
Gauge_theory
Harmonic functions as solutions to Laplace's equation
subgroup of the conformal group as functions on a multiply connected manifold or orbifold. From the fact that the group of conformal transforms is infinite-dimensional
Potential_theory
Study of angle-preserving transformations
since they are non-conformal (see below). Möbius group elements are analytic functions of the whole plane and so are necessarily conformal. Consider, in the
Inversive_geometry
Model of the extended complex plane plus a point at infinity
The Riemann surface's conformal structure does, however, determine a class of metrics: all those whose subordinate conformal structure is the given one
Riemann_sphere
Description of gravity using discrete values
dynamics, MOND AQUAL Tensor–vector–scalar Nonsymmetric gravitation Scalar–tensor theories Brans–Dicke Scalar–tensor–vector Conformal gravity Scalar theories
Quantum_gravity
Upper-half plane model of hyperbolic non-Euclidean geometry
to a null vector, which can also be thought of as a kind of stereographic projection centered on an ideal point. The projection is conformal, meaning that
Poincaré_half-plane_model
Element of an exterior algebra
algebra Λ(V) of a vector space V. This algebra is graded, associative and alternating, and consists of linear combinations of simple k-vectors (also known as
Multivector
Locally spherical point on a mathematical surface
are equal, hence, both principal curvatures are equal, and every tangent vector is a principal direction. The name "umbilic" comes from the Latin umbilicus
Umbilical_point
Mechanism that explains the generation of mass for gauge bosons
coordinates in a complex two dimensional vector space. Rotating the coordinates so that the second basis vector points in the direction of the Higgs boson
Higgs_mechanism
Formulation of classical mechanics
point particles with masses m1, m2, ..., mN, each particle has a position vector, denoted r1, r2, ..., rN. Cartesian coordinates are often sufficient, so
Lagrangian_mechanics
Geographic data file
size in the x-direction in map units/pixel Line 2: D: rotation about y-axis Line 3: B: rotation about x-axis Line 4: E: pixel size in the y-direction
World_file
∞ ( R n ) {\displaystyle C_{0}^{\infty }(\mathbf {R} ^{n})} and their conformal equivalents on the sphere, the Laplacian in euclidean n-space and the
Clifford_analysis
Physical theory describing classical fields
assigning a vector to each point in space. Each vector represents the direction of the movement of air at that point, so the set of all wind vectors in an area
Classical_field_theory
Theorem of gravity in cosmology
{\displaystyle I_{BB}={\frac {3\pi c^{3}}{\Lambda G\hbar ln2}}.} Within the conformal cyclic cosmology this theorem implies that, in each aeon of an initial
Gurzadyan_theorem
Mathematical space used to study hyperbolic geometry
vector spaces are used in Euclidean geometry. Ungar introduced the concept of gyrovectors that have addition based on gyrogroups instead of vectors which
Gyrovector_space
Type of 2D conformal field theory
fact that the corresponding WZW models are logarithmic conformal field theories. The known conformal field theories based on affine Lie algebras are not
Wess–Zumino–Witten_model
Transformations induced by a mathematical group
counterclockwise rotation through an angle α about an axis given by a unit vector v; z is the same rotation; see quaternions and spatial rotation. This is not
Group_action
Isometry group of Euclidean space
orthogonal group. The Euclidean group E(n) comprises all translations, rotations, and reflections of E n {\displaystyle \mathbb {E} ^{n}} ; and arbitrary
Euclidean_group
Regular object in four dimensional geometry
the laws of physics. Cambridge University Press. Dorst, Leo (2019). "Conformal Villarceau Rotors". Advances in Applied Clifford Algebras. 29 (44) 44
24-cell
Unified field theory
four dimensions of space and time; a 4-vector A μ {\displaystyle A^{\mu }} identified with the electromagnetic vector potential; and a scalar field ϕ {\displaystyle
Kaluza–Klein_theory
Main-belt asteroid
rotation while monitoring its brightness over the two months leading up to its encounter. The obliquity of Donaldjohanson's angular momentum vector is
52246_Donaldjohanson
These are nonlinear conformal ("angle preserving") transformations. One has Lorentz transformations ⊂ Poincaré transformations ⊂ conformal group transformations
Derivations of the Lorentz transformations
Derivations_of_the_Lorentz_transformations
Construction analogous to that of a dual vector space
lattices, the dual lattice is a construction analogous to that of a dual vector space. In certain respects, the geometry of the dual lattice of a lattice
Dual_lattice
Geometric algebra approach to gravity
first argument and a is a constant vector. Similarly, a rotation by some arbitrary rotor R gives rise to the rotation-gauge field Ω ¯ ( a , x ) ↦ Ω ¯ ′
Gauge_theory_gravity
Geometric property of some molecules and ions
having chiral gauche conformers that belong to the C2 point group, butane is considered achiral at room temperature because rotation about the central C–C
Chirality_(chemistry)
CONFORMAL ROTATION-VECTOR
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CONFORMAL ROTATION-VECTOR
CONFORMAL ROTATION-VECTOR
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CONFORMAL ROTATION-VECTOR