About a geometrical curve, a conic section the term used in rhetoric Hyperbole File Hyperbola PSF .png right thumb 250px A hyperbola is an open curve with two branches, the intersection of a plane geometry ... is the hyperbola. A double cone consists of two cones stacked point to point and sharing the same axis .... In mathematics a hyperbola is a curve, specifically a smooth function smooth curve that lies ... for which it is the solution set. A hyperbola has two pieces, called connected components or branches, which are mirror images of each other and resembling two infinite bow weapon bows . The hyperbola ... the line on the cone and the axis, or if the plane is parallel to the axis, then the conic is a hyperbola ... instead of attractive forces but the principle is the same , and so on. Each branch of the hyperbola consists of two arms which become straighter lower curvature further out from the center of the hyperbola ... of symmetry of the hyperbola, which can be thought of as the mirror point about which each branch ... of sign in some term. Many other mathematical object s have their origin in the hyperbola, such as hyperbolic ... . History The word hyperbola derives from the Greek language Greek polytonic , meaning over thrown or excessive , from which the English term hyperbole also derives. The term hyperbola is believed ... hyperbola , less than one ellipse and exactly one parabola , respectively. Nomenclature File Hyperbola properties.svg right frame The asymptotes of the hyperbola red curves are shown as blue dashed lines and intersect at the center of the hyperbola, C . The two focal points are labeled F sub 1 sub ... 2 sub . The eccentricity e equals the ratio of the distances from a point P on the hyperbola to one ... with the transverse axis. Similar to a parabola , a hyperbola is an open curve, meaning that it continues indefinitely to infinity, rather than closing on itself as an ellipse does. A hyperbola consists ... diameter of an ellipse. The midpoint of the transverse axis is known as the hyperbola s center . The distance ... more details
File Drini conjugatehyperbolas.svg thumb right The Unit Hyperbola is blue, its conjugate is green, and the asymptotes are red. In geometry , the unit hyperbola is the set of points x,y in the Cartesian plane that satisfies math x 2 y 2 1 . math In the study of pseudo Euclidean space s and indefinite orthogonal group s, the unit hyperbola forms the basis for an alternative radial length math r sqrt x 2 y 2 . math Whereas the unit circle surrounds its center, the unit hyperbola requires the conjugate hyperbola math y 2 x 2 1 math to complement it in the plane. This pair of hyperbolas share the asymptote s y x and y &minus x . When the conjugate of the unit hyperbola is in use, the alternative radial ... transformations the unit hyperbola, its conjugate hyperbola, the axes of the hyperbola, a diameter of the unit hyperbola, and the conjugate diameters conjugate diameter . The plane with the axes refers to a resting frame of reference . The diameter of the unit hyperbola represents a frame of reference ... on the unit hyperbola. The conjugate diameter represents the spatial hyperplane of simultaneity corresponding to rapidity a . In this context the unit hyperbola is a calibration hyperbola ref Anthony ... main hyperbolic angle As a particular conic section conic , the hyperbola can be parametrized by the process ... to AB intersects the conic a second time to be the sum of the points A and B . For the hyperbola ... hyperbola is key to the split complex number plane consisting of z x y j where j sup 2 sup 1 ... swaps the unit hyperbola with its conjugate, and takes any hyperbola diameter to the conjugate diameter. In terms of the hyperbolic angle parameter a , the unit hyperbola consists of points math pm cosh a j sinh a math where j 0,1 . The right branch of the unit hyperbola corresponds to the positive .... Unlike the circle group , this unit hyperbola group is not compact space compact . Similar to the ordinary ... drawn from the parametrization of the unit hyperbola and the alternative radial length. References ... more details
Discovery The nine point hyperbola was first discovered by E.F. Allen and his work was published in a volume of The American Mathematical Monthly in December, 1941. Allen was able to take the work that English Mathematician Frank Morley completed on the nine point circle using complex numbers in his book Inverse Geometry 1933 and apply it to hyperbolas using split complex numbers using the equation zz 1 for hyperbolas in the split complex plane. The nine point hyperbola was recalled by Isaak Yaglom when describing Minkowskian geometry in the conclusion of his book A Simple Non Euclidean Geometry and its Physical Basis 1979 . For Yaglom, a hyperbola is a Minkowskian circle . He says on page 193 ...the midpoints of the sides of a triangle ABC and the feet of its altitudes as well as the midpoints of the segments joining the orthocenter of ABC to its vertices lie on a Minkowskian circle S whose radius is half the radius of the circumcircle of the triangle. It is natural to refer to S as the six nine point circle of the Minkowskian triangle ABC if the triangle ABC has an incircle s , then the six nine point circle S of ABC touches its incircle s Fig.173 . Construction File Hyperbola.jpg thumb left Hyperbola Beginning the construction. Starting out with a right hyperbola, , we can find ..., we use a drafting compass to find two alternate points on the right hyperbola. When you draw a line through one of those points and the constructed foci point you get another point on the hyperbola ... given a rectangular hyperbola, this same nine point circle can be constructed using the same methods used above when complete the rectangular nine point hyperbola would look like this. See also Nine Point Circle Hyperbolas References Allen, E.F. On a Triangle Inscribed in a Rectangular Hyperbola, The American ... http mathworld.wolfram.com RectangularHyperbola.html Mathworld s Rectangular Hyperbola DEFAULTSORT Nine Point Hyperbola Category Triangle geometry geometry stub ... 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 Hyperbolic function Hyperbolic geometry Hyperbolic growth Hyperbolic paraboloid not to be confused with hyperboloid Hyperbolic manifold Hyperbolic space Hyperbolic trajectory disambig ... more details
A nodoid is a surface of revolution with constant nonzero mean curvature obtained by rolling a hyperbola along a fixed line, tracing the Focus geometry focus , and revolving the resulting curve around the line. geometry stub Category Surfaces ... more details
File Smoothed Octagon Simple.svg thumb 150px A smoothed octagon. File Smoothed Octagon Packed.svg thumb 150px The best known packing of smoothed octagons. The smoothed octagon is a geometrical construction conjectured to have the lowest maximum packing density of the Plane geometry plane of all centrally symmetric convex shapes. It is constructed by replacing the corners of a regular octagon with a section of a hyperbola that is tangent to the two sides adjacent to the corner and asymptotic to the sides adjacent to these. The smoothed octagon has a maximum packing density, sub so sub given by math eta so frac 8 4 sqrt 2 ln 2 2 sqrt 2 1 approx 0.902414 , . math ref MathWorld urlname SmoothedOctagon title Smoothed Octagon ref This is lower than the circle packing maximum packing density of circles , which is math frac pi sqrt 12 approx 0.9069. math Construction File Smoothed Octagon.svg thumb 400px Construction of the smoothed octagon black , the tangent hyperbola red and the asymptotes of this hyperbola green , and the tangent sides to the hyperbola blue . The hyperbola is constructed tangent to two sides of the octagon, and asymptotic to the two adjacent to these. If we define two constants, and m math ell sqrt 2 1 math math m sqrt 6 sqrt 2 8 frac sqrt 2 1 2 math The hyperbola is then given by the equation math ell 2x 2 y 2 m 2 math or the equivalent parametrisation for the right hand branch only math x frac m ell cosh t quad y m sinh t quad pi t pi math The lines of the octagon tangent to the hyperbola are math y pm left sqrt 2 1 right left x 2 right math The lines asymptotic to the hyperbola are simply math y pm ell x. math See also Circle packing References Reflist External links http www.home.unix ag.org scholl octagon.html The thinnest densest two dimensional packing? . Peter Scholl, 2001. Category Tiling ... more details
Unreferenced date December 2009 In geometry , confocal means having the same Focus geometry foci . For an optical cavity consisting of two mirrors, confocal means that they share their foci. If they are identical mirrors, their radius of curvature , R sub mirror sub , equals L , where L is the distance between the mirrors. In conic section s, it is said of two ellipse s, two hyperbola s, or an ellipse and a hyperbola which share both foci with each other. If an ellipse and a hyperbola are confocal, they are perpendicular to each other. In optics , it means that one Focus optics focus or image point of one lens is the same as one focus of the next lens optics lens . See also Confocal laser scanning microscopy Confocal microscopy Category Elementary geometry Category Optics de Konfokal eo Samfokusa it Confocale ... more details
Orphan date November 2006 In Reflection seismology , stacking velocity is the value of velocity obtained from the best fit hyperbola analysis. References http www.glossary.oilfield.slb.com Display.cfm?Term stacking 20velocity Schlumberger Oilfield Glossary DEFAULTSORT Stacking Velocity Category Geology geol stub ... more details
Transverse axis refers to an axis which is wikt Transverse transverse side to side, relative to some defined forward direction . In particular Transverse axis aircraft For a hyperbola the transverse axis is in the same direction as the semi major axis . disambig Long comment to prevent listing on Special Shortpages.......................................................................... more details
than b . Hyperbola In a hyperbola, a conjugate axis or minor axis of length 2 b , corresponding ... connecting the two vertices turning points of the hyperbola, with the two axes intersecting at the center of the hyperbola. The endpoints 0, b of the minor axis lie at the height of the asymptotes over under the hyperbola s vertices. Either half of the minor axis is called the semi minor axis ... minor and semi major axes lengths appear in the equation of the hyperbola relative to these axes ... through the eccentricity, as follows math b a sqrt e 2 1 . math Note that in a hyperbola b ... more details
The Dupin indicatrix is a method for characterising the local shape of a surface . Draw a plane parallel to the tangent plane and a small distance away from it. Consider the intersection of the surface with this plane. The shape of the intersection is related to the Gaussian curvature . The Dupin indicatrix is the result of the limiting process as the plane approaches the tangent plane. The indicatrix was invented by Charles Dupin . For elliptical points where the Gaussian curvature is positive the intersection will either be empty or form a closed curve. In the limit this curve will form an ellipse aligned with the principal curvature principal direction s. For hyperbolic points, where the Gaussian curvature is negative, the intersection will form a hyperbola . Two different hyperbola will be formed on either side of the tangent plane. These hyperbola share the same axis and asymptotes. The directions of the asymptotes are the same as the asymptotic direction s. See also Euler s theorem differential geometry References citation last Eisenhart first Luther P. authorlink Luther Eisenhart title A Treatise on the Differential Geometry of Curves and Surfaces publisher Dover year 2004 id ISBN 0486438201 http www.archive.org details treatonthediffer00eiserich Full 1909 text now out of copyright geometry stub Category Differential geometry of surfaces Category Surfaces de Indikatrix ru ... more details
Polar distance may refer to Polar distance astronomy , an astronomical term associated with the celestial equatorial coordinate system , ellipse and lower, a hyperbola Polar distance geometry , more correctly called Radial distance geometry radial distance , typically denoted r , a coordinate in polar coordinate system s r , Polar distance botany is used in the classification of pollen s disambig ... more details
Summary Author FlightGlobal Source http www.flightglobal.com blogs hyperbola 2009 02 picture indias manned orbital.html URL http www.flightglobal.com blogs hyperbola ISRO 20orbital 20vehicle.jpg Licensing Non free use rationale Description Image from FlightGlobal website Source http www.flightglobal.com blogs hyperbola 2009 02 picture indias manned orbital.html Article ISRO Orbital Vehicle Portion All Low resolution No Purpose To illustrate ISRO OV Replaceability other information Non free fair use in ISRO OV ... more details
x 1 t 1 math . Given a hyperbola with asymptote A , its reflection in A produces the conjugate hyperbola . Any diameter of the original hyperbola is reflected to a conjugate diameters conjugate diameter .... As E. T. Whittaker wrote in 1910, the hyperbola is unaltered when any pair of conjugate diameters ... more details
You may be looking for Efficient Frontier company Image markowitz frontier.jpg right frame Efficient Frontier. The hyperbola is sometimes referred to as the Markowitz Bullet , and its upward sloped portion is the efficient frontier if no risk free asset is available. With a risk free asset, the straight line is the efficient frontier. The efficient frontier is a concept in Modern portfolio theory introduced by Harry Markowitz and others. A combination of assets, i.e. a portfolio finance portfolio , is referred to as efficient if it has the best possible expected value expected level of return finance return for its level of risk usually proxied by the standard deviation of the portfolio s return . Here, every possible combination of risky assets, without including any holdings of the Risk free interest rate risk free asset , can be plotted in risk expected return space, and the collection of all such possible portfolios defines a region in this space. The left boundary of this region, a hyperbola , is then called the Efficient Frontier . See Modern portfolio theory The efficient frontier with no risk free asset further under Modern portfolio theory . See also Modern portfolio theory External links http www.investopedia.com terms e efficientfrontier.asp Efficient Frontier , investopedia http www.riskglossary.com link efficient frontier.htm Efficient Frontier , riskglossary.com Econ theory stub Category Financial economics Category Finance theories Category Mathematical finance Category Portfolio theories ... more details
Unreferenced date December 2009 Year nav topic 1668 science The year 1668 in science and technology involved some significant events. Astronomy Isaac Newton invents the reflecting telescope. Biology Francesco Redi disproves theories of the spontaneous generation of maggot s in putrefying matter. Mathematics Nicholas Mercator and William Brouncker discover an infinite series for the logarithm while attempting to calculate the area under a hyperbola hyperbolic segment . Births Hermann Boerhaave , Netherlands Dutch physician and chemist who made Leiden a Europe an centre of medical knowledge died 1738 in science 1738 Deaths Category 1668 in science fr 1668 en science ... more details
About the term used in rhetoric the mathematical term Hyperbola Wiktionary hyperbole Hyperbole pron en ha p rb li respell hy PUR b lee ref http www.oxfordadvancedlearnersdictionary.com dictionary hyperbole The Oxford Advanced Learner s Dictionary ref from ancient Greek polytonic exaggeration is the use of exaggeration as a rhetorical device or figure of speech . It may be used to evoke strong feelings or to create a strong impression, but is not meant to be taken literally. Hyperboles are exaggerations to create emphasis or effect. As a literary device , hyperbole is often used in poetry , and is frequently encountered in casual speech. An example of hyperbole is The bag weighed a ton. ref cite book last Mahony first David title Literacy Tests Year 7 year 2003 publisher Pascal Press isbn 9781877085369 page 82 ref Hyperbole helps to make the point that the bag was very heavy, although it is not probable that it would actually weigh a ton. In rhetoric , some opposites of hyperbole are meiosis figure of speech meiosis , litotes , understatement , and bathos the letdown after a hyperbole in a phrase . References Reflist Category Rhetorical techniques bs Hiperbola figura bg ca Hip rbole cs Hyperbola literatura cy Gormodiaith de Hyperbel Sprache es Hip rbole eo Troigo eu Hiperbole fa fr Hyperbole rh torique gl Hip rbole hr Hiperbola figura id Hiperbol is kjur it Iperbole figura retorica he ka lt Hiperbol menas mk nl Hyperbool stijlfiguur ja no Hyperbol pl Hiperbola teoria literatury pt Hip rbole figura de estilo ro Hiperbol figur de stil ru simple Exaggeration sk Hyperbola literat ra sq Hiperbola sr sh Hiperbola figura fi Hype sv Hyperbol tr Abart c l k uk zh yue zh ... more details
Hyperbolic motion is the motion of an object with constant proper acceleration in special relativity . It is called hyperbolic motion because the equation describing the path of the object through spacetime is a hyperbola , as can be seen when graphed on a Minkowski diagram . The proper acceleration of a particle is defined as the acceleration that a particle feels as it accelerates from one inertial reference frame to another. This can be derived mathematically as math alpha frac 1 left 1 u 2 c 2 right 3 2 frac du dt math , where math u math is the instantaneous speed of the particle. Solving for the equation of motion results in math x 2 c 2t 2 c 4 alpha 2 math , which is a hyperbola. Hyperbolic motion is easily visualized on a Minkowski diagram, where the motion of the accelerating particle is along the math x math axis. Each hyperbola is defined by math X c 2 alpha math . Image HyperbolicMotion.PNG See also Rindler coordinates References Ludwik Silberstein 1914 List of publications in physics The Theory of Relativity The Theory of Relativity , page 190. Naber, Gregory L., The Geometry of Minkowski Spacetime , Springer Verlag, New York, 1992. ISBN 0 387 97848 8 hardcover , ISBN 0 486 43235 1 Dover paperback edition . pp 58 60. External links http math.ucr.edu home baez physics Relativity SR rocket.html The Relativistic Rocket, John Baez, UC Riverside Category Relativity it Moto iperbolico relativit ... more details
of only one dimension of space, the other generated by time. In such a plane, one hyperbola corresponds to events a constant distance from the origin event, the other hyperbola corresponds to events ... more details
jesuit Alphonse Antonio de Sarasa was a Jesuit mathematician who contributed to the understanding of logarithm s, particularly as area s under a hyperbola . Alphonse de Sarasa was born in 1618, in Nieveport in Flanders. In 1632 he was admitted as a novice in Ghent . It was there that he worked alongside Gregoire de Saint Vincent whose ideas he developed, exploited, and promulgated. According to Sommervogel 1896 , Alphonse de Sarasa also held academic positions in Antwerp and Brussels. In 1649 Alphonse de Sarasa published Solutio problematis a R.P. Marino Mersenne Minimo propositi . This book was in response to Marin Mersenne s pamphlet Reflexiones Physico mathematicae which reviewed Saint Vincent s Opus Geometricum and posed this challenge Given three arbitrary magnitudes, rational or irrational, and given the logarithms of the two, to find the logarithm of the third geometrically. R.P. Burn 2001 explains that the term logarithm was used differently in the seventeenth century. They were any arithmetic progression which corresponded to a geometric progression . Burn says, in reviewing de Sarasa s popularization of de Saint Vincent, and concurring with Moritz Cantor , that the relationship between logarithms and the hyperbola was found by Saint Vincent in all but name . Burn quotes de Sarasa on this point the foundation of the teaching embracing logarithms are contained in Saint Vincent s Opus Geometricum , part 4 of Book 6, de Hyperbola . Alphonse Antonio de Sarasa died in Brussels in 1667. See also List of Roman Catholic scientist clerics References R. P. Burn 2001 Alphonse Antonio de Sarasa and Logarithms , Historia Mathematica 28 1 17. C. Sommervogel 1896 Biblioth que de la Compagnie de J sus , vol. VII, pp. 621 7. Use dmy dates date January 2011 Persondata Metadata see Wikipedia Persondata . NAME Sarasa, Alphonse Antonio de ALTERNATIVE NAMES SHORT DESCRIPTION Jesuit mathematician DATE OF BIRTH 1618 PLACE OF BIRTH Nieveport, Flanders DATE OF DEATH 1667 PLACE ... more details
and if e 1 the conic is a hyperbola. If the distance to the focus is fixed and the directrix ... becomes a closed curve elliptical projection . To generate a hyperbola, the radius of the directrix ... thus, the focus is outside the directrix circle. The arms of the hyperbola approach asymptotic lines and the right hand arm of one branch of a hyperbola meets the left hand arm of the other branch of a hyperbola at the point at infinity this is based on the principle that, in projective geometry, a single line meets itself at a point at infinity. The two branches of a hyperbola are thus the two ... 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
Unreferenced date December 2009 Polar distance PD is an astronomy astronomical term associated with the celestial equatorial coordinate system , and it is an angular distance of a celestial object on its meridian astronomy meridian measured from the celestial pole , similar as declination dec, is measured from the celestial equator Definition Polar distace PD 90 Polar distances are expressed in degree angle degree s and cannot exceed 90 in magnitude. An object on the celestial equator has a PD of 90 . Polar distance is not affected by the precession of the equinoxes . If the polar distance of the Sun is equal to the observer s latitude , the shadow path of a gnomon s tip on a sundial will be a parabola at higher latitudes it will be an ellipse and lower, a hyperbola . DEFAULTSORT Polar Distance Astronomy Category Celestial coordinate system Category Angle el ... more details
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