Euclidean orthogonality is preserved by rotation in the left diagram; hyperbolic orthogonality with respect to hyperbola (B) is preserved by hyperbolic rotation in the right diagram.
In geometry, the relation of hyperbolic orthogonality between two lines separated by the asymptotes of a hyperbola is a concept used in special relativity to define simultaneous events. Two events will be simultaneous when they are on a line hyperbolically orthogonal to a particular time line. This dependence on a certain time line is determined by velocity, and is the basis for the relativity of simultaneity.
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In geometry, the relation of hyperbolicorthogonality between two lines separated by the asymptotes of a hyperbola is a concept used in special relativity...
In mathematics, orthogonality is the generalization of the geometric notion of perpendicularity. Orthogonality is also used with various meanings that...
clock Hyperbolic number, a synonym for split-complex number Hyperbolicorthogonality, an orthogonality found in pseudo-Euclidean space Hyperbolic paraboloid...
rapidity, and the ordinary sense of simultaneity becomes dependent on hyperbolicorthogonality of spatial directions to the worldline associated to the rapidity...
relativity Fundamental concepts Frame of reference Speed of light Hyperbolicorthogonality Rapidity Maxwell's equations Proper length Proper time Proper acceleration...
defined, or the basis for Minkowski geometry through the notion of hyperbolicorthogonality. In this viewpoint, an affine transformation is a projective transformation...
In mathematics, hyperbolic geometry (also called Lobachevskian geometry or Bolyai–Lobachevskian geometry) is a non-Euclidean geometry. The parallel postulate...
light being infinite. Mathematically, rapidity can be defined as the hyperbolic angle that differentiates two frames of reference in relative motion,...
To uphold causality, Minkowski restricted spacetime to non-Euclidean hyperbolic geometry.[page needed] Because signals and other causal influences cannot...
cone are self-orthogonal. When a time event and a space event evaluate to zero under the bilinear form, then they are hyperbolic-orthogonal. This terminology...
diameters Conjugate diameters – Perpendicular diameters of a circle or hyperbolic-orthogonal diameters of a hyperbola Diameter (group theory) – measure of a...
be expressed as a logarithmic function or, equivalently, as an inverse hyperbolic function: τ ( t ) = c g ln ( g t c + 1 + ( g t c ) 2 ) = c g arsinh...
relativity Fundamental concepts Frame of reference Speed of light Hyperbolicorthogonality Rapidity Maxwell's equations Proper length Proper time Proper acceleration...
squeeze mappings as hyperbolic rotation is analogous to interpreting the group SO(2) (the connected component of the definite orthogonal group) preserving...
with analytic geometry. Orthogonality of electric and magnetic vector fields in space was extended by hyperbolicorthogonality for the temporal factor...
perpendicularity is the relation of conjugate diameters of a circle, so hyperbolicorthogonality is the relation of conjugate diameters of rectangular hyperbolas...
relativity Fundamental concepts Frame of reference Speed of light Hyperbolicorthogonality Rapidity Maxwell's equations Proper length Proper time Proper acceleration...
relativity Fundamental concepts Frame of reference Speed of light Hyperbolicorthogonality Rapidity Maxwell's equations Proper length Proper time Proper acceleration...
CAT(k) space. Consequently, its fundamental group Γ = π1(M) is Gromov hyperbolic. This has many implications for the structure of the fundamental group:...
versa), see hyperbolicorthogonality. A vector e is called a unit vector if η(e, e) = ±1. A basis for M consisting of mutually orthogonal unit vectors...
relativity Fundamental concepts Frame of reference Speed of light Hyperbolicorthogonality Rapidity Maxwell's equations Proper length Proper time Proper acceleration...
ISBN 978-0-8218-3900-3. Arlan Ramsay; Robert D. Richtmyer (1995). Introduction to Hyperbolic Geometry. Springer. p. 11. ISBN 0-387-94339-0. geometry axiom coordinate...
When fully written out, the EFE are a system of ten coupled, nonlinear, hyperbolic-elliptic partial differential equations. The above form of the EFE is...
relativity Fundamental concepts Frame of reference Speed of light Hyperbolicorthogonality Rapidity Maxwell's equations Proper length Proper time Proper acceleration...
relativity Fundamental concepts Frame of reference Speed of light Hyperbolicorthogonality Rapidity Maxwell's equations Proper length Proper time Proper acceleration...
are orthogonal generalized circles. In the conformal disk model of the hyperbolic plane, every geodesic is an arc of a generalized circle orthogonal to...
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