, hyperbolic functions are analogs of the ordinary trigonometric function trigonometric , or circular, functions. The basic hyperbolic functions are the hyperbolic sine sinh typically pron en s nt or IPA en a n , and the hyperbolic cosine cosh typically pron en k , from which are derived the hyperbolic tangent tanh typically pron en t nt or IPA en n , and so on, corresponding to the derived trigonometric functions. The inverse hyperbolicfunction s are the area hyperbolic ... when used with positive exponents. Another notation for the hyperbolic cotangent function is ctnh ... . The Gudermannian function gives a direct relationship between the circular functions and the hyperbolic ... uniform gravity. Relationship to the exponential function File Hyperbolic and exponential cosh.png ... ix , math Thus, hyperbolic functions are periodic function periodic with respect to the imaginary component ... of hyperbolic functions Poinsot s spirals Sigmoid function References references External links ... DEFAULTSORT HyperbolicFunction Category Elementary special functions Category Exponentials ...Image Hyperbolic functions 2.svg thumb 296px right A ray through the origin intercepts the hyperbola ... the right half of the equilateral hyperbola . Hyperbolic functions occur in the solutions of some ... electromagnetic theory , heat transfer , fluid dynamics , and special relativity . The hyperbolic functions take real values for a real argument called a hyperbolic angle . In complex analysis, they are simply rational function s of exponential function exponentials , and so are meromorphic function meromorphic . Hyperbolic functions were introduced in the 1760s independently by Vincenzo Riccati ... to refer to hyperbolic functions. Lambert adopted the names but altered the abbreviations to what they are today. ref Georg F. Becker. Hyperbolic functions. Read Books, 1931. Page xlviii. ref Standard ... csch font , font color 00b300 sech font and font color 0000b3 coth font The hyperbolic functions ... more details
Image Area tangent.svg thumb 200px right The function artanh . The inverse function inverse s of the hyperbolicfunction s are the area hyperbolic functions . The names hint at the fact that they give the area of a hyperbolic sector sector of the unit hyperbola nowrap 1 x sup 2 sup &minus y sup 2 sup 1 in the same way that the inverse trigonometric function s give the arclength of a sector on the unit circle nowrap 1 x sup 2 sup y sup 2 sup 1 . The abbreviations arcsinh, arccosh, etc., are commonly ... Oxford Oxford University Press , 2004 , ISBN 0198507631, Section 0.2.13 The inverse hyperbolic functions , p. 68 The Latin names for the inverse hyperbolic functions are area sinus hyperbolicus ... inverse hyperbolic functions. ref ref name BronshteinEtAl2007 As stated by Ilja N. Bronshtein, Konstantin ... 2 , Section 2.10 Area Functions , p. 91 blockquote The area functions are the inverse functions of the hyperbolic functions, i.e., the inverse hyperbolic functions . The functions sinh x , tanh x , and coth x are strictly monotone, so they have unique inverses without any restriction the function cosh ... that the geometric definition of the functions is the area of certain hyperbolic sectors ... blockquote ... e.g., cosh sup &minus 1 sup x versus cosh x sup &minus 1 sup . The values of inverse hyperbolic functions are hyperbolic angle s. Logarithmic representation The operators are defined in the complex ... true when using principal square roots. style text align center Inverse hyperbolic functions ... frac 1 cosh theta frac 1 sqrt 1 sinh 2 theta frac 1 sqrt 1 x 2 math Composition of hyperbolic and inverse hyperbolic functions math begin align & operatorname sinh operatorname arcosh ,x sqrt x 2 ... left v sqrt 1 u 2 u sqrt v 2 1 right end align math See also List of integrals of inverse hyperbolic ... Inverse hyperbolic functions at MathWorld http www.ucl.ac.uk Mathematics geomath level2 hyper hy8.html Inverse hyperbolic functions at University College London Department of Mathematics Category ... more details
wiktionary Hyperbolic refers to something related to or in shape of hyperbola a type of curve , or to something employing the literary device of hyperbole overstatement or plausible exaggeration . The following topics are based on the hyperbola etymology HyperbolicfunctionHyperbolic geometry Hyperbolic growth Hyperbolic paraboloid not to be confused with hyperboloid Hyperbolic manifold Hyperbolic space Hyperbolic trajectory disambig ... more details
In mathematics, hyperbolic trigonometry can mean The use of the hyperbolicfunction s The use of gyrotrigonometry in hyperbolic geometry Mathdab ... more details
plane bounded by three tangent circles. Euclidean geometry In the foundations of the hyperbolicfunction s sinh, cosh and tanh, a hyperbolic triangle is a right triangle in the first quadrant ...In mathematics , the term hyperbolic triangle has more than one meaning. Image Uniform tiling 73 t2.png thumb right 200px A tiling of the hyperbolic plane with hyperbolic triangles &ndash the order 7 triangular tiling . Hyperbolic geometry In hyperbolic geometry , a hyperbolic triangle is a figure in the hyperbolic ... in this unit. In a hyperbolic triangle the sum of the angles A , B , C respectively opposite to the side ... called the defect geometry defect of the triangle. The area of a hyperbolic triangle is equal ... If C is a right angle then The sine of angle A is the ratio of the hyperbolic sine of the side opposite the angle to the hyperbolic sine of the hypotenuse . math sin A frac textrm sinh opposite textrm sinh hypotenuse frac sinh a , sinh c , . , math The cosine of angle A is the ratio of the hyperbolic tangent of the adjacent leg to the hyperbolic tangent of the hypotenuse. math cos A frac textrm tanh ... of the hyperbolic tangent of the opposite leg to the hyperbolic sine of the adjacent leg. math tan A frac textrm tanh opposite textrm sinh adjacent frac tanh a , sinh b , . , math The hyperbolic sine, cosine, and tangent are hyperbolic functions which are analogous to the standard trigonometric functions. Oblique triangles Whether C is a right angle or not, the following relationships hold. There is a hyperbolic ... three are vertices are ideal, then the resulting figure is called an ideal triangle . An ideal hyperbolic ... is math sqrt 2 sinh a, , math where a is the appropriate hyperbolic angle . See also Hyperbolic ... of common and hyperbolic trigonometry . citation first Wilson last Stothers title Hyperbolic geometry url http www.maths.gla.ac.uk wws cabripages hyperbolic hyperbolic0.html publisher University of Glasgow ... in chapter 1. DEFAULTSORT Hyperbolic Triangle Category Hyperbolic geometry it Triangolo iperbolico ... more details
Circle inversion inversion in unit semicircle . The general hyperbolic motion is a function composition composition of fundamental hyperbolic motions. Elementary half plane geometry Consider the triangle ... rays by the logarithm function center d x , y , x , z log z y . center Then by means of hyperbolic ...for hyperbolic motion in physics hyperbolic motion relativity In geometry , a hyperbolic motion is a mapping of a model of hyperbolic geometry that preserves the distance measure in the model. Such a mapping ... of rotation s and translation s. One uses hyperbolic motions to relate structures within the model. The collection of all hyperbolic motions form a group mathematics group which characterizes the geometry according to the Erlangen program . Hyperbolic motions are visualized in the upper half plane ... fundamental hyperbolic motions p q x c , y , c R left or right shift p q sx , sy , s 0 dilation ... is sec a see secant . Set r sec a and apply the third fundamental hyperbolic motion to obtain ... ,0 . Thus the tangent ray at 1,0 gets mapped to Z by the third fundamental hyperbolic motion ... it to the tangent ray. So the collection of hyperbolic motions permutes the semicircles with diameters ... the hyperbolic lines in HP. Since the erection of the HP model relies deeply on Euclidean geometry and traditional trigonometry especially tangent and secant , it is natural to consider hyperbolic geometry ... to the boundary of D signifying hyperbolic lines . Using the arithmetic and geometry of complex numbers, and Mobius transformation s, there is the Poincar disc model of the hyperbolic plane ... the hyperbolic lines we see that these transformations are motions of the D model of hyperbolic ... sphere in the ring mathematics ring of coquaternion s. References Lars Ahlfors 1967 Hyperbolic ... from euclidean surfaces to hyperbolic knots , Chapter 2 The Hyperbolic Plane , pages 11&ndash ... , Mir Publishers , Moscow. Category Inversive geometry Category Hyperbolic geometry ... more details
Image Hyperbolic sector.svg 200px right A hyperbolic angle in standard position is the angle at 0, 0 between the ray to 1, 1 and the ray to x , 1 x where x 1. The magnitude of the hyperbolic angle is the area of the corresponding hyperbolic sector which is log sub e sub x . Note that unlike circular angle, hyperbolic angle is unbounded , as is the function natural logarithm log sub e sub x , a fact related to the unbounded nature of the harmonic series mathematics harmonic series . The hyperbolic angle is considered to be negative when 0 x 1. The hyperbolicfunction s sinh, cosh, and tanh use the hyperbolic ... i x i sinh x math so that the hyperbolicfunction s cosh and sinh can be presented through the circular ... of holomorphic function s, the hyperbolic sine and cosine functions are incorporated into the complex ... may be premised on analogies to circular trigonometric functions when the hyperbolic angle defines a hyperbolic triangle . Thus this parameter becomes one of the most useful in the calculus of a real ... Circular vs hyperbolic angle click to see animation Hyperbolic angles can be motivated by considering ... reciprocal slopes hyperbolic orthogonality . Whereas in Euclidean geometry moving steadily ... distances upon the hyperbolic line. ref http www.math.cornell.edu web4520 CG15 0.pdf Hyperbolic Geometry ... . It thus makes sense to define the hyperbolic angle from P sub 0 sub to an arbitrary point on the curve as a logarithmic function of the point s value of x . ref Bj rn Felsager, http www.chartwellyorke.com ... a point on a hyperbolic sector sweeps out an area proportional to the hyperbolic angle. A circle centred ... 2 math . The same area units can also be used to give a scale to hyperbolic angles. It is notable that the hyperbola through 1,1 is always as far or further from the origin than the circle, so the hyperbolic ... the topic of hyperbolic angle. The quadrature of the hyperbola was first accomplished by Gregoire ... of Mathematics , pp. 424,5 v. 1 The upshot was the logarithm function, as now understood as the area ... more details
cleanup date August 2008 Image Rectangular hyperbola.svg thumb The reciprocal function , exhibiting hyperbolic ... exhibits hyperbolic growth with a singularity at time math t c math in the limit of a function limit ... to the square of the value of x in the moment t. Respectively, the quadratic hyperbolicfunction ... variation it is said to undergo hyperbolic growth . ref See, e.g., Korotayev A., Malkov A., Khaltourina .... P. 19 20. ref Is any singularity called by analogy hyperbolic growth? 1 x 2 has a singularity at 0, but the graph is not a hyperbola. More precisely, the reciprocal function math 1 x math has a hyperbola as a graph, and has a singularity at 0, meaning that the limit of a function limit as math x to 0 math is infinity any similar graph is said to exhibit hyperbolic growth. Description If the output of a function is inversely proportional to its input, or inversely proportional to the difference from a given value math x 0 math , the function will exhibit hyperbolic growth, with a singularity at math x 0 math . In the real world hyperbolic growth is created by certain non linear positive feedback mechanisms. Comparisons with other growth Like exponential growth and logistic growth , hyperbolic ... can be confused, as exponential growth, hyperbolic growth, and the first half of logistic growth are convex function s however their asymptotic behavior behavior as input gets large differs dramatically ... growth grows to infinity as time goes to infinity but is always finite for finite time , hyperbolic ... Certain mathematical models suggest that until the early 1970s the world population underwent hyperbolic ... the hyperbolic growth of the world population was accompanied by quadratic hyperbolic growth of the world .... The hyperbolic growth of the world population and quadratic hyperbolic growth of the world GDP ..., logistic growth, or other functions. Queuing theory Another example of hyperbolic growth can be found ... as a function of the average load ratio of the server. The singularity in this case occurs when the average ... more details
Image Hyperbolic orthogonal dodecahedral honeycomb.png thumb A perspective projection of a Hyperbolic small dodecahedral honeycomb dodecahedral tessellation in Hyperbolic 3 manifold H sup 3 sup . BR Four ... cubic tessellation in Euclidean space E sup 3 sup In mathematics , hyperbolic space is a type of non Euclidean geometry . Whereas spherical geometry has a constant positive curvature, hyperbolic geometry has a negative curvature every point in hyperbolic space is a saddle point . Parallel lines ... is the amount of space covered by the n ball in hyperbolic n space it increases exponentially with respect to the radius of the ball, rather than polynomially. Formal definition Hyperbolic n space ... with constant sectional curvature &minus 1. Hyperbolic space is the principal example of a space exhibiting hyperbolic geometry . It can be thought of as the negative curvature analogue of the n sphere . Although hyperbolic space H sup n sup is diffeomorphic to R sup n sup its negative curvature metric gives it very different geometric properties. Hyperbolic 2 space, H , is also called the hyperbolic plane. Models of hyperbolic space Hyperbolic space, developed independently by Lobachevsky ... the hyperbolic plane uniquely up to isometry there is an extra constant, the curvature K ... length scale, one can thus assume, without loss of generality, that K 1. Hyperbolic spaces are constructed ... geometry. There are several important models of hyperbolic space the Klein model , the hyperboloid ... hyperbolic space as a hyperboloid in R sup n 1 sup x sub 0 sub ,..., x sub n sub x sub i sub ... instead of hyperbolic functions. ref the distance between two points x and y on H to be math d x, y operatorname arccosh , B x,y . math This function satisfies the axioms of a metric space . Moreover ... An alternative model of hyperbolic geometry is on a certain open set domain in projective space . The Minkowski ... for which Q x 0 in the homogeneous coordinates x . The domain U sup n sup is the Klein model of hyperbolic ... more details
math . This is a continuous mapping , but not an analytic function . Quadrant model of hyperbolic geometry The correspondence math Q leftrightarrow HP math affords the hyperbolic geometry structure to Q that is erected on HP by hyperbolic motion s. The hyperbolic lines in Q are Line mathematics Ray ...Original research date June 2010 Image Hyperbolic coordinates.svg thumb 400px right Hyperbolic coordinates plotted on the Cartesian plane u in blue and v in red. In mathematics , hyperbolic coordinates ... 0, y 0 Q math . Hyperbolic coordinates take values in math HP u, v u in mathbb R , v 0 math . These coordinates ... frac y x right math and math v sqrt xy math . Sometimes the parameter math u math is called hyperbolic ... of the hyperbolic plane. Indeed, consider rays from the origin in Q , and their images, vertical ... follows the hyperbolic path and work thermodynamics work can be interpreted as a hyperbolic angle ... in the quadrant of absolute temperature and gas density. For hyperbolic coordinates in the Theory ... in the quadrant. Economic applications There are many natural applications of hyperbolic coordinates ... hyperbolic angle. For a fluctuation take a new price math 0 z y math . Then the change in u ... hyperbolic angle provides an objective, symmetric, and consistent measure mathematics measure . The quantity math Delta u math is the length of the left right shift in the hyperbolic motion view of the currency .... History While the geometric mean is an ancient concept, the hyperbolic angle is contemporary with the development ... , Marin Mersenne , and Alphonse Antonio de Sarasa evaluated the quadrature of the hyperbola as a function having properties now familiar for the logarithm. The exponential function, the hyperbolic sine, and the hyperbolic cosine followed. As complex function theory referred to infinite series the circular functions sine and cosine seemed to absorb the hyperbolic sine and cosine as depending on an imaginary ... complex plane called split complex number s where the hyperbolic angle is raised to a level equal ... more details
all measured in this unit then math cosh c cosh a cosh b ,. math The cosh function is a hyperbolic ...Image Hyperbolic.svg frame right Lines through a given point P and asymptotic to line R . FAIR USE of Hyperbolic parallels.gif see image description page at http en.wikipedia.org wiki Image Hyperbolic parallels.gif for rationale Image with unknown copyright status removed Image Hyperbolic parallels.gif frame right hyperbolic lines that violate the parallel postulate Image Hyperbolic triangle.svg thumb 250px right A triangle immersed in a saddle shape plane a hyperbolic paraboloid , as well as two diverging ultraparallel lines. In mathematics , hyperbolic geometry also called Nikolai Lobachevsky ... to l . In hyperbolic geometry there are at least two distinct lines through P which do not intersect l , so the parallel postulate is false. Models of the hyperbolic plane Models have been constructed within Euclidean geometry that obey the axioms of hyperbolic geometry, thus proving that the parallel postulate is independent of the other postulates of Euclid. Because there is no precise hyperbolic analogue to Euclidean parallel lines, the hyperbolic use of parallel and related terms varies ... property of hyperbolic geometry is that the angles of a hyperbolic triangle Hyperbolic geometry ... ideal triangle ideal hyperbolic triangles in which all three angles are 0 . Non intersecting lines An interesting property of hyperbolic geometry follows from the occurrence of more than one ..., any smaller angle will force the line to intersect l . This is called an asymptotic line in hyperbolic ... postulate In hyperbolic geometry, given any line l , and point P not on l , there are exactly ... theorem states that there is a unique line in the hyperbolic plane that is perpendicular ... equal to 90 . In hyperbolic geometry, the angle of parallelism varies with the math Pi p math function. This function, described by Nikolai Ivanovich Lobachevsky , produces a unique angle of parallelism ... more details
a hyperbolicfunctionhyperbolic shape , and that this shape should produce a reversal of preference ... Quasi hyperbolic approximation The quasi hyperbolic discount function, proposed by Laibson 1997 , ref name Laibson1997QJE approximates the hyperbolic discount function above in discrete time by math ...Refimprove date June 2010 In behavioral economics , hyperbolic discounting is a time inconsistent model ... A Critical Review. Journal of Economic Literature . Vol. 40, No. 2, pp. 351 401 ref Hyperbolic .... Hyperbolic discounting has been observed in humans and animals. In hyperbolic discounting, valuations ... s hyperbolic discounting curve is to compare short term preferences with long term preferences. For instance ... biased preferences present biased . Individuals using hyperbolic discounting reveal a strong ... Laibson1997QJE Laibson, David, 1997. Golden Eggs and Hyperbolic Discounting, The Quarterly Journal of Economics, MIT Press, vol. 112 2 , pages 443 77, May. ref happens because hyperbolic discounts value future rewards much more than exponential discounting. Observations The phenomenon of hyperbolic ... have confirmed that spontaneous preferences by both human and nonhuman subjects follow a hyperbolic ... that is the same choice seen at five years greater distance. Hyperbolic discounting has also been ... of hyperbolic discounting to find that drug dependent individuals discount delayed consequences ..., 5 , 256 262. ref ref Vuchinich, R.E., and Simpson, C.A. 1998 . Hyperbolic temporal discounting ... high rates of hyperbolic discounting precede addictions or vice versa is currently unknown, although ... rate hyperbolic discounting makes unpredictable gambling outcomes more satisfying. ref Madden, G.J. ... is vitally important in describing hyperbolic discounting, especially in the discounting of specific ... Hyperbolic vs. exponential discount factors.svg thumb 300px right Comparison of the discount factors of hyperbolic and exponential discounting. In both cases, math k 1 math . Hyperbolic discounting ... more details
Probability distribution name hyperbolic type density pdf image cdf image parameters math mu math location parameter location real number real br math alpha math to do real br math beta math asymmetry parameter real br math delta math scale parameter real br math gamma sqrt alpha 2 beta 2 math support math x in infty infty math pdf math frac gamma 2 alpha delta K 1 delta gamma e alpha sqrt delta 2 x mu 2 beta x mu math br br math K lambda math denotes a modified Bessel function of the second kind cdf to do mean math mu frac delta beta K 2 delta gamma gamma K 1 delta gamma math median to do mode math mu frac delta beta gamma math variance math frac delta K 2 delta gamma gamma K 1 delta gamma frac beta 2 delta 2 gamma 2 left frac K 3 delta gamma K 1 delta gamma frac K 2 2 delta gamma K 1 2 delta gamma right math skewness to do kurtosis to do entropy to do mgf math frac e mu z gamma K 1 delta alpha 2 beta z 2 alpha 2 beta z 2 K 1 delta gamma math char to do The hyperbolic distribution is a continuous probability distribution that is characterized by the fact that the logarithm of the probability density function is a hyperbola . Thus the distribution decreases exponentially, which is more slowly than the normal distribution . It is therefore suitable to model phenomena where numerically large values are more probable than is the case for the normal distribution. Examples are returns from financial assets and turbulent wind speeds. The hyperbolic distributions form a subclass of the generalised hyperbolic distribution s. The origin of the distribution is the observation by Ralph Alger Bagnold in his book The Physics of Blown Sand and Desert Dunes 1941 that the logarithm of the histogram ... stable 79167 publisher The Royal Society ref , where he also introduced the generalised hyperbolic distribution , using the fact the a hyperbolic distribution is a random mixture of normal distributions. References references DEFAULTSORT Hyperbolic Distribution Category Continuous distributions ... more details
In mathematics, the term hyperbolic plane may refer to A two dimensional plane in hyperbolic geometry A two dimensional plane in Minkowski space the plane of the hyperbolic number s A two dimensional quadratic space with a non singular isotropic quadratic form mathdab zh ... more details
In mathematics , a hyperbolic link is a link knot theory link in the 3 sphere with knot complement complement that has a complete Riemannian metric of constant negative curvature , i.e. has a hyperbolic geometry . A hyperbolic knot is a hyperbolic link with one connected space component . As a consequence of the work of William Thurston , it is known that every knot is precisely one of the following hyperbolic, a torus knot , or a satellite knot . As a consequence, hyperbolic knots can be considered plentiful. A similar heuristic applies to hyperbolic links. As a consequence of Thurston s hyperbolic Dehn surgery theorem, performing Dehn surgery Dehn surgeries on a hyperbolic link enables one to obtain many more hyperbolic 3 manifold s. Examples File BorromeanRings.svg thumb Borromean rings are a hyperbolic link. Borromean rings are hyperbolic. Every split link non split , prime knot prime , alternating knot alternating link that is not a torus link is hyperbolic by a result of William Menasco . See also SnapPea hyperbolic volume knot References Colin Adams, The Knot Book , American Mathematical Society, ISBN 0 8050 7380 9 William Menasco, Closed incompressible surfaces in alternating knot and link complements . Topology 23 1984 , no. 1, 37 44. William Thurston, The geometry and topology of 3 manifolds , Princeton lecture notes 1978 1981 . Further reading Colin Adams, http front.math.ucdavis.edu math.GT 0309466 Hyperbolic knots arXiv preprint Category Knot theory Category 3 manifolds topology stub ... more details
Image Hyperbolic sector.svg 200px right A hyperbolic sector is a region of the Cartesian plane x , y bounded by rays from the origin to two points a , 1 a and b , 1 b and by the hyperbola xy 1. A hyperbolic sector in standard position has a 1 and b 1 . The area of a hyperbolic sector in standard position is natural logarithm log sub e sub b . Proof Integrate under 1 x from 1 to b , add triangle 0, 0 , 1, 0 , 1, 1 , and subtract triangle 0, 0 , b , 0 , b , 1 b When in standard position, a hyperbolic sector corresponds to a positive hyperbolic angle . See also Squeeze mapping geometry stub Category Curves Category Elementary geometry ar bs Hiperboli ki sektor es Sector hiperb lico pt Setor hiperb lico zh ... more details
Hyperbolic structure may refer to Hyperboloid structure Hyperbolic set mathdab Short pages monitor This long comment was added to the page to prevent it being listed on Special Shortpages. It and the accompanying monitoring template were generated via Template Longcomment. Please do not remove the monitor template without removing the comment as well. ... more details
Orphan date February 2009 In applied mathematics , a hyperbolic point in a system dx dt F x of ordinary differential equation s is a stationary point x sub 0 sub such that the eigenvalue s of the linearized system have non zero real part. See also Anticlastic Elliptic point Gaussian curvature Hyperbolic fixed point Parabolic point Planar point Synclastic External links http mathworld.wolfram.com HyperbolicPoint.html Hyperbolic Point at Wolfram MathWorld Mathapplied stub Category Differential geometry of surfaces ... more details
In group theory , a hyperbolic group , also known as a word hyperbolic group , Gromov hyperbolic group ... satisfying certain properties characteristic of hyperbolic geometry . The notion of a hyperbolic .... He noticed that many results of Max Dehn concerning the fundamental group of a hyperbolic Riemann ... program. Ideas and foundational material in the theory of hyperbolic groups also stem from the work ... Hyperbolic groups can be defined in several different ways. All definitions use the Cayley graph of the group and involve a choice of a positive constant and first define a hyperbolic group . A group is called hyperbolic if it is hyperbolic for some . When translating between different definitions of hyperbolicity, the particular value of may change, but the resulting notions of a hyperbolic ... graph T is hyperbolic if all geodesic triangles are slim, and in this case G is a hyperbolic ... and hence to a different condition for G to be hyperbolic, it is known that the notion of hyperbolicity ... a hyperbolic group depends only on the group itself. Remark By imposing the slim triangles condition on geodesic metric spaces in general, one arrives at the more general notion of hyperbolic space math delta math hyperbolic space . Hyperbolic groups can be characterized as groups G which admit an isometric properly discontinuous action on a proper geodesic hyperbolic space X such that the factor space X G has finite diameter. Examples of hyperbolic groups Finite group s. Virtually cyclic group ... are hyperbolic, namely, the fundamental group s of surfaces with negative Euler characteristic . For example ... two is a hyperbolic group. Most triangle group s math Delta l,m,n math are hyperbolic, namely, those ... ones as special cases. It also leads to many examples of hyperbolic groups not related to trees or manifolds. Examples of non hyperbolic groups The free abelian group free rank 2 abelian group Z sup 2 sup is not hyperbolic. More generally, any group which contains Z sup 2 sup as a subgroup is not hyperbolic ... more details
A hyperbolic 3 manifold is a 3 manifold equipped with a complete space complete Riemannian metric of constant sectional curvature 1. In other words, it is the quotient of three dimensional hyperbolic space by a subgroup of hyperbolic isometries acting freely and Properly discontinuous action properly discontinuously . See also Kleinian model . Its thick thin decomposition has a thin part consisting ... The first cusped hyperbolic 3 manifold to be discovered was the Gieseking manifold , in 1912. It is constructed by gluing faces of an ideal hyperbolic tetrahedron together. The complements of knot mathematics knots and knot mathematics links in the 3 sphere are frequently cusped hyperbolic ... is neither a satellite knot nor a torus knot is a hyperbolic knot. William Thurston Thurston s theorem on hyperbolic Dehn surgery states that, provided a finite collection of filling slopes are avoided, the remaining Dehn filling s on hyperbolic links are hyperbolic 3 manifolds. The Seifert Weber space is a compact hyperbolic 3 manifold, obtained by gluing opposite faces of a dodecahedron together. The Weeks manifold has the smallest volume of any closed orientable hyperbolic 3 manifold. Thurston gave a necessary and sufficient criterion for a surface bundle over the circle to be hyperbolic ..., any closed, irreducible , atoroidal 3 manifold with infinite fundamental group is hyperbolic. There is an analogous statement for 3 manifolds with boundary. See also Hyperbolic manifold Kleinian group Mostow rigidity theorem Arithmetic hyperbolic 3 manifold References Citation last1 Maclachlan first1 Colin last2 Reid first2 Alan W. title The arithmetic of hyperbolic 3 manifolds url http books.google.com ... first1 John G. title Foundations of hyperbolic manifolds origyear 1994 publisher Springer Verlag ..., Kleinian groups and hyperbolic geometry url http dx.doi.org 10.1090 S0273 0979 1982 15003 0 doi .... New Series issn 0002 9904 volume 6 issue 3 pages 357 381 Category 3 manifolds Category Hyperbolic ... more details
In mathematics , a hyperbolic n manifold is a complete Riemannian manifold Riemannian n manifold of constant sectional curvature 1. Every complete, connected, simply connected manifold of constant negative curvature 1 is isometric to the real hyperbolic space H sup n sup . As a result, the universal cover of any closed manifold M of constant negative curvature 1 is H sup n sup . Thus, every such M can be written as H sup n sup where is a torsion free discrete group of isometries on H sup n sup . That is, is a lattice in SO su p b 1, n R . Its thick thin decomposition has a thin part consisting of tubular neighborhoods of closed geodesics and ends which are the product of a Euclidean n 1 manifold and the closed half ray. The manifold is of finite volume if and only if its thick part is compact. For n 2 the hyperbolic structure on a finite volume hyperbolic n manifold is unique by Mostow rigidity theorem Mostow rigidity and so geometric invariants are in fact topological invariants. See also Hyperbolic 3 manifold Margulis lemma Hyperbolic space Hyperbolization theorem References Citation last1 Kapovich first1 Michael title Hyperbolic manifolds and discrete groups origyear 2001 url http dx.doi.org 10.1007 978 0 8176 4913 5 publisher Birkh user Boston location Boston, MA series Modern Birkh user Classics isbn 978 0 8176 4912 8 doi 10.1007 978 0 8176 4913 5 id MR 1792613 year 2009 Citation last1 Maclachlan first1 Colin last2 Reid first2 Alan W. title The arithmetic of hyperbolic 3 manifolds url http books.google.com books?id yrmT56mpw3kC publisher Springer Verlag location Berlin, New York series Graduate Texts in Mathematics isbn 978 0 387 98386 8 id MR 1937957 year 2003 volume 219 Citation last1 Ratcliffe first1 John G. title Foundations of hyperbolic manifolds origyear 1994 publisher Springer Verlag location Berlin, New York edition 2nd series Graduate Texts in Mathematics ... Hyperbolic geometry Category Manifolds it Variet iperbolica ... more details
In Web development jargon and Visualization graphic information visualization , a hyperbolic tree often shortened as hypertree defines a visualization method for a Graph mathematics graph inspired by hyperbolic geometry . Image BasicTree.png thumb right 256px A basic hyperbolic tree. Nodes in focus are placed in the center and given more room, while out of focus nodes are compressed near the boundaries. Image BasicTreeFocused.png thumb right 256px Focusing on a different node brings it and its children to the center of the disk, while uninteresting portions of the tree are compressed. Displaying hierarchical data as a Tree data structure tree suffers from visual clutter as the number of nodes per level can grow exponentially. For a simple binary tree, the maximum number of nodes at a level ... is to use a hyperbolic tree , first introduced by Lamping et al. ref http sigchi.org chi95 Electronic documnts papers jl bdy.htm ref Hyperbolic trees employ hyperbolic geometry hyperbolic space , which ... in Euclidean space increases its circumference linearly, while the same circle in hyperbolic ... out the tree in hyperbolic space in an uncluttered manner placing a node far enough from its parent ... a hyperbolic tree commonly utilizes the Poincar disk model Poincare disk model of hyperbolic geometry, though the Klein model Klein Beltrami model can also be used. Both display the entire hyperbolic ... out of focus closer to the boundary of the disk. Traversing the hyperbolic tree requires M bius ... out of view. Although hyperbolic trees have been patented in the U.S. by Xerox, various Java & JavaScript ... 2FPTO 2Fsearch bool.html&r 1&f G&l 50&d PALL&RefSrch yes&Query PN 2F5590250 ref See also Hyperbolic ... jl bdy.htm http thejit.org JavaScript InfoVis Toolkit has an interactive Hyperbolic Tree visualization ... last3 Pirolli first3 Peter title A Focus Context Technique Based on Hyperbolic Geometry for Visualizing ... div references DEFAULTSORT Hyperbolic Tree Category Hyperbolic geometry Category Visualization graphic ... more details
File OrbitalEccentricityDemo.svg thumb right The blue path in this image is an example of a hyperbolic trajectory. In astrodynamics or celestial mechanics a hyperbolic trajectory is a Kepler orbit with the Orbital eccentricity eccentricity greater than 1. Under standard assumptions a body traveling along this trajectory will orbital coast coast to infinity, arriving there with hyperbolic excess velocity relative to the central body . Similarly to parabolic trajectory all hyperbolic trajectories are also escape trajectory escape trajectories . The specific orbital energy specific energy of a hyperbolic trajectory orbit is positive. Hyperbolic excess velocity seealso Characteristic energy Under standard assumptions the body traveling along hyperbolic trajectory will attain in infinity an orbital velocity called hyperbolic excess velocity math v infty , math that can be computed as math v infty sqrt mu over a , math where math mu Gm , math is standard gravitational parameter , math a , math is the negative semi major axis of orbit s hyperbola . The hyperbolic excess velocity is related to the specific orbital energy or characteristic energy by math 2 epsilon C 3 v infty 2 , math Velocity Under standard assumptions the orbital velocity math v , math of a body traveling along hyperbolic trajectory can be computed as Vallado math v sqrt mu left 2 over r 1 over a right math where math ... velocity math v esc , math and hyperbolic excess velocity math v infty , math math v 2 v esc 2 v infty ... orbit eccentricity , which is greater than 1 for hyperbolic trajectories. Distance of closest approach ... , math of a hyperbolic trajectory is greater than zero and the orbital energy conservation equation ... parameter . Radial hyperbolic trajectory A radial hyperbolic trajectory is a non periodic Radial .... This is a hyperbolic orbit with semi minor axis 0 and eccentricity 1. Although the eccentricity ... iperbolica ja pt Trajet ria hiperb lica simple Hyperbolic trajectory tr Hiperbolik y r nge ... more details
In dynamical systems theory , a subset &Lambda of a smooth manifold M is said to have a hyperbolic structure with respect to a smooth map f if its tangent bundle may be split into two invariant subbundle s, one of which is contracting and the other is expanding under f , with respect to some Riemannian metric on M . An analogous definition applies to the case of flow mathematics flows . In the special case when the entire manifold M is hyperbolic, the map f is called an Anosov diffeomorphism . The dynamics of f on a hyperbolic set, or hyperbolic dynamics , exhibits features of local structural stability and has been much studied, cf Axiom A . Definition Let M be a compact space compact smooth manifold , f M &rarr M a diffeomorphism , and Df TM &rarr TM the pushforward differential differential of f . An f invariant subset &Lambda of M is said to be hyperbolic , or to have a hyperbolic structure , if the restriction to &Lambda of the tangent bundle of M admits a splitting into a Whitney sum of two Df invariant subbundles, called the stable bundle and the unstable bundle and denoted E sup s sup and E sup u sup . With respect to some Riemannian metric on M , the restriction of Df to E sup s sup must be a contraction and the restriction of Df to E sup u sup must be an expansion. Thus, there exist constants 0 &lambda 1 and c 0 such that math T Lambda M E s oplus E u math and math Df x E s x E s f x math and math Df x E u x E u f x math for all math x in Lambda math and math Df nv le c lambda n v math for all math v in E s math and math n 0 math and math Df n v le c lambda n v math for all math v in E u math and math n 0 math . If &Lambda is hyperbolic then there exists a Riemannian metric for which c 1 such a metric is called adapted . Examples Hyperbolic equilibrium point p ... with period n is hyperbolic if and only if Df sup n sup at any point of the orbit has no eigenvalue ... id 4338 title Hyperbolic Set Category Dynamical systems Category Limit sets ru ... more details
class wikitable align right style text align center Hyperbolic quaternion multiplication width 15 width ... of algebra over a field algebras over a field , the hyperbolic quaternion math q a bi cj dk, quad ... 1. The four dimensional algebra of hyperbolic quaternions incorporates some of the features of the older ... as a union of complex planes union of complex planes , so the hyperbolic quaternion algebra is a union ... of lectures at Lehigh University in 1900 see Historical Review below . The hyperbolic ... kinematics by suggesting a mathematical model When one selects a unit vector r in the hyperbolic quaternions, then r sup 2 sup 1. The plane math D r lbrace t x r t, x in R rbrace math with hyperbolic ... number plane. The versor Hyperbolic versor hyperbolic versor math exp a r cosh a r sinh a math ... a t x cosh a r . math Since the direction r in space is arbitrary, this hyperbolic quaternion multiplication can express any Lorentz boost using the parameter a called rapidity . However, the hyperbolic ... 1967 191 ref Algebraic structure Like the quaternions , the set of hyperbolic quaternions form a vector space over the real numbers of dimension 4. A linear combination math q a bi cj dk math is a hyperbolic ... j 2 k 2 math Unlike the ordinary quaternions, the hyperbolic quaternions are not associative . For example ... subplane of the set M of hyperbolic quaternions that contains the real axis forms a plane of split ... of the real part of the hyperbolic quaternion product pq math p 0q 0 p 1q 1 p 2q 2 p 3q 3 math . Note ... of the Royal Society at Edinburgh in 1900. In it he treats a model for hyperbolic space H sup 3 sup ... and consists of all the versor Hyperbolic versor hyperbolic versors in the ring of hyperbolic quaternions ... parameter group is the versor Hyperbolic versor hyperbolic versor with the hyperbolic angle ... the hyperbolic quaternion ring different The basis math 1, ,i, ,j, ,k math of the vector space of hyperbolic ... with. So the appearance of Macfarlane s hyperbolic quaternions had some motivation, but the disagreeable ... more details
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