About a geometric curve the term used in rhetoric Hyperbole File Hyperbola PSF .png right thumb 210px A hyperbola is an open curve with two branches, the intersection of a plane geometry plane with both ... off parts of the top and bottom half the boundary curve of the slice on the cone is the hyperbola. A double ... holder. Each cone of light is drawing a branch of a hyperbola on a nearby vertical wall. In mathematics a hyperbola is a curve, specifically a smooth function smooth curve that lies in a plane, which ... set. A hyperbola has two pieces, called connected component graph theory connected component ... . The hyperbola is one of the four kinds of conic section , formed by the intersection of a plane ... is a hyperbola. Hyperbolas arise in practice in many ways as the curve representing the function ... of the hyperbola consists of two arms which become straighter lower curvature further out from the center of the hyperbola. Diagonally opposite arms one from each branch tend in the limit to a common ... is at the center of symmetry of the hyperbola, which can be thought of as the mirror point about ... more than a change of sign in some term. Many other mathematical object s have their origin in the hyperbola ... which is not Euclidean . History The word hyperbola derives from the Greek language Greek lang .... The term hyperbola is believed to have been coined by Apollonius of Perga ca. 262 BC ca. 190 BC ... is greater than one hyperbola , less than one ellipse and exactly one parabola , respectively. Nomenclature and features 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 ... from a point P on the hyperbola to one focus and its corresponding directrix line shown in green . The two ... sub , and br angle formed by each asymptote with the transverse axis. Similar to a parabola , a hyperbola ... as an ellipse does. A hyperbola consists of two disconnected curve s called its arms or branches ... 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 ... 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 ... . When the conjugate of the unit hyperbola is in use, the alternative radial length is math r sqrt y 2 x 2 . math The unit hyperbola finds applications where the circle must be replaced with the hyperbola ... Euclidean space . There the asymptotes of the unit hyperbola form a light cone . Further, the attention ... parametrization of the hyperbola by sector areas. When the notions of conjugate hyperbolas ... the unit circle, can be replaced with numbers built around the unit hyperbola. Minkowski diagram main ... Minkowski used to describe the relativity transformations the unit hyperbola, its conjugate hyperbola, the axes of the hyperbola, a diameter of the unit hyperbola, and the conjugate diameters conjugate ... hyperbola represents a frame of reference in motion with rapidity a where math tanh a y x math and x,y is the endpoint of the diameter 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 French 1968 Special Relativity , page 83, W. W. Norton & Company ... Butterworths ref Commonly in relativity study the hyperbola with vertical axis is taken as primary ... conic section conic , the hyperbola can be parametrized by the process of addition of points ... a second time to be the sum of the points A and B . For the hyperbola math x 2 y 2 1 math with the fixed ... unit hyperbola in Elements of Dynamic 1878 by W. K. Clifford . He describes quasi harmonic motion in a hyperbola as follows The motion math rho alpha cosh nt epsilon beta sinh nt epsilon math ... 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 ... 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 ... 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 group Hyperbolic growth Hyperbolic paraboloid not to be confused with hyperboloid Hyperbolic manifold Hyperbolic space Hyperboloid structure Hyperbolic trajectory disambiguation az Hiperbola ... 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 Tessellation ... 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
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 optics 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
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 ... minor and semi major axes lengths appear in the equation of the hyperbola relative to these axes ... are related through the eccentricity, as follows math b a sqrt e 2 1 . math Note that in a hyperbola ... 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 Geology 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
Friedrich Wilhelm August Ludwig Kiepert 1846 1934 was a German mathematician who introduced the Kiepert hyperbola . References http nonio.mat.uc.pt PENSAS EN02 experdescgeomet articulos kiepert.html Friedrich Wilhelm August Ludwig Kiepert MathGenealogy id id 9714 Category German mathematicians Category 1934 deaths ... more details
In differential geometry , 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 Category Differential geometry of surfaces Category Surfaces differential geometry stub 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
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 . ref V.G. Ashkinuse & Isaak Yaglom 1962 Ideas and Methods of Affine and Projective Geometry in Russian language Russian , page 151, Ministry of Education, Moscow ref When in standard position, a hyperbolic sector corresponds to a positive hyperbolic angle . Hyperbolic logarithm Image hyperbola E.svg thumb Unit area for x e Students of integral calculus know that f x x sup p sup has an algebraic antiderivative except in the case p &minus 1 corresponding to the quadrature of the hyperbola. The other cases are given by Cavalieri s quadrature formula . Whereas quadrature of the parabola had been accomplished by Archimedes in the 3rd century BC The Quadrature of the Parabola , the hyperbolic quadrature required the invention of a new function Gregoire de Saint Vincent addressed the problem of computing the area of a hyperbolic sector. His findings led to the natural logarithm function, once called the hyperbolic logarithm since it is obtained by integrating, or finding the area, under the hyperbola. The natural logarithm is a transcendental function , an entity beyond the class of algebraic function s. Evidently transcendental functions are necessary in integral calculus. See also Squeeze mapping References reflist Category Area Category Elementary geometry Category Integral calculus ar bs Hiperboli ki sektor es Sector hiperb lico pt Setor hiperb lico zh ... more details
Harmonic motion can mean The motion of a Harmonic oscillator in physics , which can be Simple harmonic motion Complex harmonic motion Keplers laws of planetary motion in physics , known as the harmonic law Quasi harmonic motion Unit hyperbola Parametrization Musica universalis in medieval astronomy , the music of the spheres Chord progression in music , harmonic progression See also Pendulum Harmonograph Circular motion disambig ... 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
time . In such a plane, one hyperbola corresponds to events a constant distance from the origin event, the other hyperbola corresponds to events a constant proper time from it. The principle of relativity ... 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 cite journal author Born, M. title Die Theorie des starren Elektrons in der Kinematik des Relativit ts Prinzipes journal Ann. Phys. year 1909 volume 30 pages 1 doi 10.1002 andp.19093351102 bibcode 1909AnP...335....1B Wikisource translation s The Theory of the Rigid Electron in the Kinematics of the Principle of Relativity The Theory of the Rigid Electron in the Kinematics of the Principle of Relativity 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 Theory of relativity Category Article Feedback 5 it Moto iperbolico ru ... more details
About the term used in rhetoric the mathematical term Hyperbola Wiktionary hyperbole Hyperbole IPAc en icon h a p r b l i respell hy PUR b lee ref cite web title Hyperbole url http oald8.oxfordlearnersdictionaries.com dictionary hyperbole publisher Oxford Advanced Learner s Dictionary accessdate February 15, 2012 ref ancient Greek Greek lang grc , 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. ref cite web title Hyperbole url http dictionary.reference.com browse hyperbole publisher Dictionary.com accessdate February 15, 2012 ref 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 978 1 877085 36 9 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 . External links http www.poetandknowit.com english definitions hyperbole examples.aspx Examples of hyperbole in poetry References Reflist Category Rhetorical techniques Category Greek loanwords bg bs Hiperbola figura 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 ... de estilo ro Hiperbol figur de stil ru sq Hiperbola simple Exaggeration sk Hyperbola literat ra sr sh Hiperbola figura fi Hyperbola sv Hyperbol tr Abart c l k ... 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
Mergeto Spaceflight discuss Talk Spaceflight Merge discussion for Point to point sub orbital spaceflight date February 2010 Point to point sub orbital spaceflight is a category of spaceflight in which a spacecraft uses a sub orbital flight for transportation. This can provide a two hour trip from London to Sydney . Today, no company offers this type of spaceflight for transportation. However, Virgin Galactic is planning to build a spaceplane called SpaceShipThree , which will offer this service in the future. ref name fg200802 http www.flightglobal.com blogs hyperbola 2008 02 spaceshipthree revealed.html SpaceShipThree revealed? , FlightGlobal Hyperbola, Rob Coppinger, 29 Feb 2008 ref See also Non stop flight Future of ultra long haul Future of ultra long haul air travel References reflist Category Spaceflight Category Suborbital spaceflight space stub ... more details
italic title Taxobox name Hesperorhipis regnum Animal ia phylum Arthropod a classis Insect a ordo Beetle Coleoptera subordo Polyphaga superfamilia Buprestoidea familia Buprestidae genus Hesperorhipis genus authority Fall, 1930 Hesperorhipis is a genus of beetle s in the family Buprestidae , containing the following species ref cite web url http www.fond4beetles.com Buprestidae WorldCat Genera Hesperorhipis.htm title Genus Hesperorhipis author Bellamy, C. L. date 2010 work A Checklist of World Buprestoidea accessdate 16 Jun 2011 ref Hesperorhipis albofasciata small Fall, 1930 small Hesperorhipis hyperbola small Knull, 1938 small Hesperorhipis jacumbae small Knull, 1954 small Hesperorhipis mirabilis small Knull, 1937 small References reflist Category Buprestoidea Buprestidae stub ... more details
Image Excentricidad.svg thumb Point F is a focus point for the red ellipse, green parabola and blue hyperbola. In geometry , the foci IPAc en icon f o s a singular focus are a pair of special points with reference to which any of a variety of curves is constructed. For example, foci can be used in defining conic section s, the four types of which are the circle , ellipse , parabola , and hyperbola . In addition, foci are used to define the Cassini oval and the Cartesian oval . Conics in projective geometry Defining conics in terms of two foci An ellipse can be defined as the Locus mathematics locus of points for each of which the sum of the distances to two given foci is a constant. A circle is the special case of an ellipse in which the two foci coincide with each other. Thus, a circle can be more simply defined as the locus of points each of which is a fixed distance from a single given focus. A circle can also be defined as the Circles of Apollonius circle of Apollonius , in terms of two different foci, as the set of points having a fixed ratio of distances to the two foci. A parabola is a limiting case of an ellipse in which one of the foci is a point at infinity . A hyperbola can be defined as the locus of points for each of which the absolute value of the difference between the distances to two given foci is a constant. Defining conics in terms of a focus and a directrix It is also possible to describe all the conic sections in terms of a single focus and a single Conic ... zero and one the conic is an ellipse if e 1 the conic is a parabola and if e 1 the conic is a hyperbola ... projection . To generate a hyperbola, the radius of the directrix circle is chosen to be less ... 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 ... at infinity. The two branches of a hyperbola are thus the two twisted halves of a curve closed over ... more details
Image Lambert Fig2.png thumb right 300px Figure 2 Hyperbola with the points math P 1 math and math ... math The points math F 1 math , math P 1 math and math P 2 math define a hyperbola going through the point ... is either on the left or on the right branch of the hyperbola depending on the sign of math A math . The semi major axis of this hyperbola is math A math and the eccentricity math E math is math frac d A math . This hyperbola is illustrated in figure 2. Relative the usual canonical coordinate system defined by the major and minor axis of the hyperbola its equation is math frac x 2 A 2 frac y 2 ... of the hyperbola as math F 1 math the difference between the distances math r 2 math to point math ... F 2 math on the other branch of the hyperbola corresponding relation is math s 1 s 2 2A quad 4 math ... 1 r 2 cos alpha 2 quad 7 math and the semi major axis with sign of the hyperbola discussed above is math A frac r 2 r 1 2 quad 8 math The eccentricity with sign for the hyperbola is math E frac d A quad ... math F 1 math relative the canonical coordinate system for the hyperbola are note that math ... math F 2 math on the other branch of the hyperbola as free parameter the x coordinate of math .... In the special case that math r 1 r 2 math or very close math A 0 math and the hyperbola with two ... more details
Other uses Vertex disambiguation Image Ellipse evolute.svg right thumb 240px An ellipse red and its evolute blue . The dots are the vertices of the curve, each corresponding to a cusp on the evolute. In the geometry of curve s, a vertex is a point of where the first derivative of curvature is zero. This is typically a local Maxima and minima maximum or minimum of curvature. Other special cases may occur, for instance when the second derivative is also zero, or when the curvature is constant. For a circle which has constant curvature, every point is a vertex. The four vertex theorem states that every closed curve must have at least four vertices. Vertices are points where the curve has Contact mathematics Contact between curves 4 point contact with the osculating circle at that point. The evolute of a curve will generically have a cusp singularity cusp when the curve has a vertex. Other, more degenerate and non stable singularities occur at higher vertices. Higher vertices generically occur in a one parameter family of curves when two ordinary vertices coalesce to form a higher vertex after which they annihilate. The symmetry set has endpoints at the cusps corresponding to the vertices, and the medial axis , a subset of the symmetry set , also has its endpoints in the cusps. If a curve is reflection symmetry bilaterally symmetric , it will have a vertex at the point or points where the axis of symmetry crosses the curve. Thus, the notion of a vertex for a curve is closely related to that of an vertex optics optical vertex , the point where an optical axis crosses a Lens optics lens surface. Vertices of a conic section A hyperbola has two vertices, one on each branch they are the closest of any two points lying on opposite branches of the hyperbola, and they lie on the principal axis. On a parabola, the sole vertex lies on the axis of symmetry. On an ellipse, two of the four vertices lie on the major axis and two lie on the minor axis. References unreferenced date Nov ... more details
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