1.
Greek alphabet
–
It is the ancestor of the Latin and Cyrillic scripts. In its classical and modern forms, the alphabet has 24 letters, Modern and Ancient Greek use different diacritics. In standard Modern Greek spelling, orthography has been simplified to the monotonic system, examples In both Ancient and Modern Greek, the letters of the Greek alphabet have fairly stable and consistent symbol-to-sound mappings, making pronunciation of words largely predictable. Ancient Greek spelling was generally near-phonemic, among consonant letters, all letters that denoted voiced plosive consonants and aspirated plosives in Ancient Greek stand for corresponding fricative sounds in Modern Greek. This leads to groups of vowel letters denoting identical sounds today. Modern Greek orthography remains true to the spellings in most of these cases. The following vowel letters and digraphs are involved in the mergers, Modern Greek speakers typically use the same, modern, in other countries, students of Ancient Greek may use a variety of conventional approximations of the historical sound system in pronouncing Ancient Greek. Several letter combinations have special conventional sound values different from those of their single components, among them are several digraphs of vowel letters that formerly represented diphthongs but are now monophthongized. In addition to the three mentioned above, there is also ⟨ου⟩, pronounced /u/, the Ancient Greek diphthongs ⟨αυ⟩, ⟨ευ⟩ and ⟨ηυ⟩ are pronounced, and respectively in voicing environments in Modern Greek. The Modern Greek consonant combinations ⟨μπ⟩ and ⟨ντ⟩ stand for and respectively, ⟨τζ⟩ stands for, in addition, both in Ancient and Modern Greek, the letter ⟨γ⟩, before another velar consonant, stands for the velar nasal, thus ⟨γγ⟩ and ⟨γκ⟩ are pronounced like English ⟨ng⟩. There are also the combinations ⟨γχ⟩ and ⟨γξ⟩ and these signs were originally designed to mark different forms of the phonological pitch accent in Ancient Greek. The letter rho, although not a vowel, also carries a rough breathing in word-initial position, if a rho was geminated within a word, the first ρ always had the smooth breathing and the second the rough breathing leading to the transiliteration rrh. The vowel letters ⟨α, η, ω⟩ carry an additional diacritic in certain words, the iota subscript. This iota represents the former offglide of what were originally long diphthongs, ⟨ᾱι, ηι, ωι⟩, another diacritic used in Greek is the diaeresis, indicating a hiatus. In 1982, a new, simplified orthography, known as monotonic, was adopted for use in Modern Greek by the Greek state. Although it is not a diacritic, the comma has a function as a silent letter in a handful of Greek words, principally distinguishing ό. There are many different methods of rendering Greek text or Greek names in the Latin script, the form in which classical Greek names are conventionally rendered in English goes back to the way Greek loanwords were incorporated into Latin in antiquity. In this system, ⟨κ⟩ is replaced with ⟨c⟩, the diphthongs ⟨αι⟩ and ⟨οι⟩ are rendered as ⟨ae⟩ and ⟨oe⟩ respectively, and ⟨ει⟩ and ⟨ου⟩ are simplified to ⟨i⟩ and ⟨u⟩ respectively
2.
Orbit
–
In physics, an orbit is the gravitationally curved path of an object around a point in space, for example the orbit of a planet about a star or a natural satellite around a planet. Normally, orbit refers to a regularly repeating path around a body, to a close approximation, planets and satellites follow elliptical orbits, with the central mass being orbited at a focal point of the ellipse, as described by Keplers laws of planetary motion. For ease of calculation, in most situations orbital motion is adequately approximated by Newtonian Mechanics, historically, the apparent motions of the planets were described by European and Arabic philosophers using the idea of celestial spheres. This model posited the existence of perfect moving spheres or rings to which the stars and it assumed the heavens were fixed apart from the motion of the spheres, and was developed without any understanding of gravity. After the planets motions were accurately measured, theoretical mechanisms such as deferent. Originally geocentric it was modified by Copernicus to place the sun at the centre to help simplify the model, the model was further challenged during the 16th century, as comets were observed traversing the spheres. The basis for the understanding of orbits was first formulated by Johannes Kepler whose results are summarised in his three laws of planetary motion. Second, he found that the speed of each planet is not constant, as had previously been thought. Third, Kepler found a relationship between the orbital properties of all the planets orbiting the Sun. For the planets, the cubes of their distances from the Sun are proportional to the squares of their orbital periods. Jupiter and Venus, for example, are respectively about 5.2 and 0.723 AU distant from the Sun, their orbital periods respectively about 11.86 and 0.615 years. The proportionality is seen by the fact that the ratio for Jupiter,5. 23/11.862, is equal to that for Venus,0. 7233/0.6152. Idealised orbits meeting these rules are known as Kepler orbits, isaac Newton demonstrated that Keplers laws were derivable from his theory of gravitation and that, in general, the orbits of bodies subject to gravity were conic sections. Newton showed that, for a pair of bodies, the sizes are in inverse proportion to their masses. Where one body is more massive than the other, it is a convenient approximation to take the center of mass as coinciding with the center of the more massive body. Lagrange developed a new approach to Newtonian mechanics emphasizing energy more than force, in a dramatic vindication of classical mechanics, in 1846 le Verrier was able to predict the position of Neptune based on unexplained perturbations in the orbit of Uranus. This led astronomers to recognize that Newtonian mechanics did not provide the highest accuracy in understanding orbits, in relativity theory, orbits follow geodesic trajectories which are usually approximated very well by the Newtonian predictions but the differences are measurable. Essentially all the evidence that can distinguish between the theories agrees with relativity theory to within experimental measurement accuracy
3.
Apsis
–
An apsis is an extreme point in an objects orbit. The word comes via Latin from Greek and is cognate with apse, for elliptic orbits about a larger body, there are two apsides, named with the prefixes peri- and ap-, or apo- added to a reference to the thing being orbited. For a body orbiting the Sun, the point of least distance is the perihelion, the terms become periastron and apastron when discussing orbits around other stars. For any satellite of Earth including the Moon the point of least distance is the perigee, for objects in Lunar orbit, the point of least distance is the pericynthion and the greatest distance the apocynthion. For any orbits around a center of mass, there are the terms pericenter and apocenter, periapsis and apoapsis are equivalent alternatives. A straight line connecting the pericenter and apocenter is the line of apsides and this is the major axis of the ellipse, its greatest diameter. For a two-body system the center of mass of the lies on this line at one of the two foci of the ellipse. When one body is larger than the other it may be taken to be at this focus. Historically, in systems, apsides were measured from the center of the Earth. In orbital mechanics, the apsis technically refers to the distance measured between the centers of mass of the central and orbiting body. However, in the case of spacecraft, the family of terms are used to refer to the orbital altitude of the spacecraft from the surface of the central body. The arithmetic mean of the two limiting distances is the length of the axis a. The geometric mean of the two distances is the length of the semi-minor axis b, the geometric mean of the two limiting speeds is −2 ε = μ a which is the speed of a body in a circular orbit whose radius is a. The words pericenter and apocenter are often seen, although periapsis/apoapsis are preferred in technical usage, various related terms are used for other celestial objects. The -gee, -helion and -astron and -galacticon forms are used in the astronomical literature when referring to the Earth, Sun, stars. The suffix -jove is occasionally used for Jupiter, while -saturnium has very rarely used in the last 50 years for Saturn. The -gee form is used as a generic closest approach to planet term instead of specifically applying to the Earth. During the Apollo program, the terms pericynthion and apocynthion were used when referring to the Moon, regarding black holes, the term peri/apomelasma was used by physicist Geoffrey A. Landis in 1998 before peri/aponigricon appeared in the scientific literature in 2002
4.
Hyperbola
–
In mathematics, a hyperbola is a type of smooth curve lying in a plane, defined by its geometric properties or by equations for which it is the solution set. A hyperbola has two pieces, called connected components or branches, that are mirror images of each other, the hyperbola is one of the three kinds of conic section, formed by the intersection of a plane and a double cone. If the plane intersects both halves of the double cone but does not pass through the apex of the cones, each branch of the hyperbola has two arms which become straighter further out from the center of the hyperbola. Diagonally opposite arms, one from each branch, tend in the limit to a common line, called the asymptote of those two arms. So there are two asymptotes, whose intersection is at the center of symmetry of the hyperbola, which can be thought of as the point about which each branch reflects to form the other branch. In the case of the curve f =1 / x the asymptotes are the two coordinate axes, hyperbolas share many of the ellipses analytical properties such as eccentricity, focus, and directrix. Typically the correspondence can be made with nothing more than a change of sign in some term, many other mathematical objects have their origin in the hyperbola, such as hyperbolic paraboloids, hyperboloids, hyperbolic geometry, hyperbolic functions, and gyrovector spaces. The word hyperbola derives from the Greek ὑπερβολή, meaning over-thrown or excessive, hyperbolae were discovered by Menaechmus in his investigations of the problem of doubling the cube, but were then called sections of obtuse cones. The term hyperbola is believed to have coined by Apollonius of Perga in his definitive work on the conic sections. The rectangle could be applied to the segment, be shorter than the segment or exceed the segment, the midpoint M of the line segment joining the foci is called the center of the hyperbola. The line through the foci is called the major axis and it contains the vertices V1, V2, which have distance a to the center. The distance c of the foci to the center is called the distance or linear eccentricity. The quotient c a is the eccentricity e, C2 is called the director circle of the hyperbola. In order to get the branch of the hyperbola, one has to use the director circle related to F1. This property should not be confused with the definition of a hyperbola with help of a directrix below, for an arbitrary point the distance to the focus is 2 + y 2 and to the second focus 2 + y 2. Hence the point is on the hyperbola if the condition is fulfilled 2 + y 2 −2 + y 2 = ±2 a. Remove the square roots by suitable squarings and use the relation b 2 = c 2 − a 2 to obtain the equation of the hyperbola, the shape parameters a, b are called the semi major axis and semi minor axis or conjugate axis. As opposed to an ellipse, a hyperbola has two vertices
5.
Geosynchronous orbit
–
A geosynchronous orbit is an orbit about the Earth of a satellite with an orbital period that matches the rotation of the Earth on its axis of approximately 23 hours 56 minutes and 4 seconds. Over the course of a day, the position in the sky traces out a path, typically in a figure-8 form, whose precise characteristics depend on the orbits inclination. Satellites are typically launched in an eastward direction, a special case of geosynchronous orbit is the geostationary orbit, which is a circular geosynchronous orbit at zero inclination. A satellite in a geostationary orbit appears stationary, always at the point in the sky. Popularly or loosely, the term geosynchronous may be used to mean geostationary, specifically, geosynchronous Earth orbit may be a synonym for geosynchronous equatorial orbit, or geostationary Earth orbit. A semi-synchronous orbit has a period of ½ sidereal day. Relative to the Earths surface it has twice this period, examples include the Molniya orbit and the orbits of the satellites in the Global Positioning System. Circular Earth geosynchronous orbits have a radius of 42,164 km, all Earth geosynchronous orbits, whether circular or elliptical, have the same semi-major axis.4418 km3/s2. In the special case of an orbit, the ground track of a satellite is a single point on the equator. A geostationary equatorial orbit is a geosynchronous orbit in the plane of the Earths equator with a radius of approximately 42,164 km. A satellite in such an orbit is at an altitude of approximately 35,786 km above sea level. It maintains the position relative to the Earths surface. The theoretical basis for this phenomenon of the sky goes back to Newtons theory of motion. In that theory, the existence of a satellite is made possible because the Earth rotates. Such orbits are useful for telecommunications satellites, a perfectly stable geostationary orbit is an ideal that can only be approximated. Elliptical geosynchronous orbits can be and are designed for satellites in order to keep the satellite within view of its assigned ground stations or receivers. A satellite in a geosynchronous orbit appears to oscillate in the sky from the viewpoint of a ground station. Satellites in highly elliptical orbits must be tracked by ground stations
6.
OCLC
–
The Online Computer Library Center is a US-based nonprofit cooperative organization dedicated to the public purposes of furthering access to the worlds information and reducing information costs. It was founded in 1967 as the Ohio College Library Center, OCLC and its member libraries cooperatively produce and maintain WorldCat, the largest online public access catalog in the world. OCLC is funded mainly by the fees that libraries have to pay for its services, the group first met on July 5,1967 on the campus of the Ohio State University to sign the articles of incorporation for the nonprofit organization. The group hired Frederick G. Kilgour, a former Yale University medical school librarian, Kilgour wished to merge the latest information storage and retrieval system of the time, the computer, with the oldest, the library. The goal of network and database was to bring libraries together to cooperatively keep track of the worlds information in order to best serve researchers and scholars. The first library to do online cataloging through OCLC was the Alden Library at Ohio University on August 26,1971 and this was the first occurrence of online cataloging by any library worldwide. Membership in OCLC is based on use of services and contribution of data, between 1967 and 1977, OCLC membership was limited to institutions in Ohio, but in 1978, a new governance structure was established that allowed institutions from other states to join. In 2002, the structure was again modified to accommodate participation from outside the United States. As OCLC expanded services in the United States outside of Ohio, it relied on establishing strategic partnerships with networks, organizations that provided training, support, by 2008, there were 15 independent United States regional service providers. OCLC networks played a key role in OCLC governance, with networks electing delegates to serve on OCLC Members Council, in early 2009, OCLC negotiated new contracts with the former networks and opened a centralized support center. OCLC provides bibliographic, abstract and full-text information to anyone, OCLC and its member libraries cooperatively produce and maintain WorldCat—the OCLC Online Union Catalog, the largest online public access catalog in the world. WorldCat has holding records from public and private libraries worldwide. org, in October 2005, the OCLC technical staff began a wiki project, WikiD, allowing readers to add commentary and structured-field information associated with any WorldCat record. The Online Computer Library Center acquired the trademark and copyrights associated with the Dewey Decimal Classification System when it bought Forest Press in 1988, a browser for books with their Dewey Decimal Classifications was available until July 2013, it was replaced by the Classify Service. S. The reference management service QuestionPoint provides libraries with tools to communicate with users and this around-the-clock reference service is provided by a cooperative of participating global libraries. OCLC has produced cards for members since 1971 with its shared online catalog. OCLC commercially sells software, e. g. CONTENTdm for managing digital collections, OCLC has been conducting research for the library community for more than 30 years. In accordance with its mission, OCLC makes its research outcomes known through various publications and these publications, including journal articles, reports, newsletters, and presentations, are available through the organizations website. The most recent publications are displayed first, and all archived resources, membership Reports – A number of significant reports on topics ranging from virtual reference in libraries to perceptions about library funding
7.
Horseshoe orbit
–
A horseshoe orbit is a type of co-orbital motion of a small orbiting body relative to a larger orbiting body. The orbital period of the body is very nearly the same as for the larger body. The loop is not closed but will drift forward or backward slightly each time, when the object approaches the larger body closely at either end of its trajectory, its apparent direction changes. Over an entire cycle the center traces the outline of a horseshoe, asteroids in horseshoe orbits with respect to Earth include 54509 YORP,2002 AA29,2010 SO16,2015 SO2 and possibly 2001 GO2. A broader definition includes 3753 Cruithne, which can be said to be in a compound and/or transition orbit, saturns moons Epimetheus and Janus occupy horseshoe orbits with respect to each other. The following explanation relates to an asteroid which is in such an orbit around the Sun, the asteroid is in almost the same solar orbit as Earth. Both take approximately one year to orbit the Sun and it is also necessary to grasp two rules of orbit dynamics, A body closer to the Sun completes an orbit more quickly than a body further away. If a body accelerates along its orbit, its orbit moves outwards from the Sun, if it decelerates, the orbital radius decreases. The horseshoe orbit arises because the attraction of the Earth changes the shape of the elliptical orbit of the asteroid. The shape changes are small but result in significant changes relative to the Earth. The horseshoe becomes apparent only when mapping the movement of the relative to both the Sun and the Earth. The asteroid always orbits the Sun in the same direction, however, it goes through a cycle of catching up with the Earth and falling behind, so that its movement relative to both the Sun and the Earth traces a shape like the outline of a horseshoe. Starting at point A, on the ring between L5 and Earth, the satellite is orbiting faster than the Earth and is on its way toward passing between the Earth and the Sun. But Earths gravity exerts an outward accelerating force, pulling the satellite into an orbit which decreases its angular speed. When the satellite gets to point B, it is traveling at the speed as Earth. Earths gravity is still accelerating the satellite along the orbital path, eventually, at Point C, the satellite reaches a high and slow enough orbit such that it starts to lag behind Earth. It then spends the next century or more appearing to drift backwards around the orbit when viewed relative to the Earth and its orbit around the Sun still takes only slightly more than one Earth year. Given enough time, the Earth and the satellite will be on opposite sides of the Sun, eventually the satellite comes around to point D where Earths gravity is now reducing the satellites orbital velocity
8.
Sine
–
In mathematics, the sine is a trigonometric function of an angle. More generally, the definition of sine can be extended to any value in terms of the length of a certain line segment in a unit circle. The function sine can be traced to the jyā and koṭi-jyā functions used in Gupta period Indian astronomy, via translation from Sanskrit to Arabic and then from Arabic to Latin. The word sine comes from a Latin mistranslation of the Arabic jiba, to define the trigonometric functions for an acute angle α, start with any right triangle that contains an angle of measure α, in the accompanying figure, angle A in triangle ABC has measure α. The three sides of the triangle are named as follows, The opposite side is the side opposite to the angle of interest, the hypotenuse is the side opposite the right angle, in this case side h. The hypotenuse is always the longest side of a right-angled triangle, the adjacent side is the remaining side, in this case side b. It forms a side of both the angle of interest and the right angle, once such a triangle is chosen, the sine of the angle is equal to the length of the opposite side divided by the length of the hypotenuse. As stated, the value sin appears to depend on the choice of right triangle containing an angle of measure α, however, this is not the case, all such triangles are similar, and so the ratio is the same for each of them. The trigonometric functions can be defined in terms of the rise, run, when the length of the line segment is 1, sine takes an angle and tells the rise. Sine takes an angle and tells the rise per unit length of the line segment, rise is equal to sin θ multiplied by the length of the line segment. In contrast, cosine is used for telling the run from the angle, arctan is used for telling the angle from the slope. The line segment is the equivalent of the hypotenuse in the right-triangle, in trigonometry, a unit circle is the circle of radius one centered at the origin in the Cartesian coordinate system. Let a line through the origin, making an angle of θ with the half of the x-axis. The x- and y-coordinates of this point of intersection are equal to cos θ and sin, the points distance from the origin is always 1. Unlike the definitions with the triangle or slope, the angle can be extended to the full set of real arguments by using the unit circle. This can also be achieved by requiring certain symmetries and that sine be a periodic function. Exact identities, These apply for all values of θ. sin = cos =1 csc The reciprocal of sine is cosecant, i. e. the reciprocal of sin is csc, or cosec. Cosecant gives the ratio of the length of the hypotenuse to the length of the opposite side, the inverse function of sine is arcsine or inverse sine
9.
Latin script
–
Latin script is used as the standard method of writing in most Western and Central European languages, as well as in many languages in other parts of the world. Latin script is the basis for the largest number of alphabets of any writing system and is the most widely adopted writing system in the world, Latin script is also the basis of the International Phonetic Alphabet. The 26 most widespread letters are the contained in the ISO basic Latin alphabet. The script is either called Roman script or Latin script, in reference to its origin in ancient Rome, in the context of transliteration, the term romanization or romanisation is often found. Unicode uses the term Latin as does the International Organization for Standardization, the numeral system is called the Roman numeral system, and the collection of the elements, Roman numerals. The numbers 1,2,3. are Latin/Roman script numbers for the Hindu–Arabic numeral system, the Latin alphabet spread, along with Latin, from the Italian Peninsula to the lands surrounding the Mediterranean Sea with the expansion of the Roman Empire. The Latin script also came into use for writing the West Slavic languages and several South Slavic languages, the speakers of East Slavic languages generally adopted Cyrillic along with Orthodox Christianity. The Serbian language uses both scripts, with Cyrillic predominating in official communication and Latin elsewhere, as determined by the Law on Official Use of the Language and Alphabet. As late as 1500, the Latin script was limited primarily to the languages spoken in Western, Northern, the Orthodox Christian Slavs of Eastern and Southeastern Europe mostly used Cyrillic, and the Greek alphabet was in use by Greek-speakers around the eastern Mediterranean. The Arabic script was widespread within Islam, both among Arabs and non-Arab nations like the Iranians, Indonesians, Malays, and Turkic peoples, most of the rest of Asia used a variety of Brahmic alphabets or the Chinese script. It is used for many Austronesian languages, including the languages of the Philippines, Latin letters served as the basis for the forms of the Cherokee syllabary developed by Sequoyah, however, the sound values are completely different. In the late 19th century, the Romanians returned to the Latin alphabet, under French rule and Portuguese missionary influence, a Latin alphabet was devised for the Vietnamese language, which had previously used Chinese characters. In 1928, as part of Mustafa Kemal Atatürks reforms, the new Republic of Turkey adopted a Latin alphabet for the Turkish language, kazakhstan, Kyrgyzstan, Iranian-speaking Tajikistan, and the breakaway region of Transnistria kept the Cyrillic alphabet, chiefly due to their close ties with Russia. In the 1930s and 1940s, the majority of Kurds replaced the Arabic script with two Latin alphabets, although the only official Kurdish government uses an Arabic alphabet for public documents, the Latin Kurdish alphabet remains widely used throughout the region by the majority of Kurdish-speakers. In 2015, the Kazakh government announced that the Latin alphabet would replace Cyrillic as the system for the Kazakh language by 2025. In the course of its use, the Latin alphabet was adapted for use in new languages, sometimes representing phonemes not found in languages that were written with the Roman characters. These new forms are given a place in the alphabet by defining an alphabetical order or collation sequence, a digraph is a pair of letters used to write one sound or a combination of sounds that does not correspond to the written letters in sequence. Examples are ⟨ch⟩, ⟨ng⟩, ⟨rh⟩, ⟨sh⟩ in English, a trigraph is made up of three letters, like the German ⟨sch⟩, the Breton ⟨c’h⟩ or the Milanese ⟨oeu⟩
10.
Geostationary orbit
–
A geostationary orbit, geostationary Earth orbit or geosynchronous equatorial orbit is a circular orbit 35,786 kilometres above the Earths equator and following the direction of the Earths rotation. An object in such an orbit has a period equal to the Earths rotational period and thus appears motionless, at a fixed position in the sky. Using this characteristic, ocean color satellites with visible and near-infrared light sensors can also be operated in geostationary orbit in order to monitor sensitive changes of ocean environments, the notion of a geostationary space station equipped with radio communication was published in 1928 by Herman Potočnik. The first appearance of an orbit in popular literature was in the first Venus Equilateral story by George O. Smith. Clarke acknowledged the connection in his introduction to The Complete Venus Equilateral, the orbit, which Clarke first described as useful for broadcast and relay communications satellites, is sometimes called the Clarke Orbit. Similarly, the Clarke Belt is the part of space about 35,786 km above sea level, in the plane of the equator, the Clarke Orbit is about 265,000 km in circumference. Most commercial communications satellites, broadcast satellites and SBAS satellites operate in geostationary orbits, a geostationary transfer orbit is used to move a satellite from low Earth orbit into a geostationary orbit. The first satellite placed into an orbit was the Syncom-3. A worldwide network of operational geostationary meteorological satellites is used to provide visible and infrared images of Earths surface, a geostationary orbit can only be achieved at an altitude very close to 35,786 km and directly above the equator. This equates to a velocity of 3.07 km/s and an orbital period of 1,436 minutes. This ensures that the satellite will match the Earths rotational period and has a footprint on the ground. All geostationary satellites have to be located on this ring. 85° per year, to correct for this orbital perturbation, regular orbital stationkeeping manoeuvres are necessary, amounting to a delta-v of approximately 50 m/s per year. A second effect to be taken into account is the longitude drift, there are two stable and two unstable equilibrium points. Any geostationary object placed between the points would be slowly accelerated towards the stable equilibrium position, causing a periodic longitude variation. The correction of this effect requires station-keeping maneuvers with a maximal delta-v of about 2 m/s per year, solar wind and radiation pressure also exert small forces on satellites, over time, these cause them to slowly drift away from their prescribed orbits. In the absence of servicing missions from the Earth or a renewable propulsion method, hall-effect thrusters, which are currently in use, have the potential to prolong the service life of a satellite by providing high-efficiency electric propulsion. This delay presents problems for latency-sensitive applications such as voice communication, geostationary satellites are directly overhead at the equator and become lower in the sky the further north or south one travels. At latitudes above about 81°, geostationary satellites are below the horizon, because of this, some Russian communication satellites have used elliptical Molniya and Tundra orbits, which have excellent visibility at high latitudes
11.
Ellipse
–
In mathematics, an ellipse is a curve in a plane surrounding two focal points such that the sum of the distances to the two focal points is constant for every point on the curve. As such, it is a generalization of a circle, which is a type of an ellipse having both focal points at the same location. The shape of an ellipse is represented by its eccentricity, which for an ellipse can be any number from 0 to arbitrarily close to, ellipses are the closed type of conic section, a plane curve resulting from the intersection of a cone by a plane. Ellipses have many similarities with the two forms of conic sections, parabolas and hyperbolas, both of which are open and unbounded. The cross section of a cylinder is an ellipse, unless the section is parallel to the axis of the cylinder and this ratio is called the eccentricity of the ellipse. Ellipses are common in physics, astronomy and engineering, for example, the orbit of each planet in our solar system is approximately an ellipse with the barycenter of the planet–Sun pair at one of the focal points. The same is true for moons orbiting planets and all other systems having two astronomical bodies, the shapes of planets and stars are often well described by ellipsoids. It is also the simplest Lissajous figure formed when the horizontal and vertical motions are sinusoids with the same frequency, a similar effect leads to elliptical polarization of light in optics. The name, ἔλλειψις, was given by Apollonius of Perga in his Conics, in order to omit the special case of a line segment, one presumes 2 a > | F1 F2 |, E =. The midpoint C of the segment joining the foci is called the center of the ellipse. The line through the foci is called the major axis and it contains the vertices V1, V2, which have distance a to the center. The distance c of the foci to the center is called the distance or linear eccentricity. The quotient c a is the eccentricity e, the case F1 = F2 yields a circle and is included. C2 is called the circle of the ellipse. This property should not be confused with the definition of an ellipse with help of a directrix below, for an arbitrary point the distance to the focus is 2 + y 2 and to the second focus 2 + y 2. Hence the point is on the ellipse if the condition is fulfilled 2 + y 2 +2 + y 2 =2 a. The shape parameters a, b are called the major axis. The points V3 =, V4 = are the co-vertices and it follows from the equation that the ellipse is symmetric with respect to both of the coordinate axes and hence symmetric with respect to the origin
12.
Fourier series
–
In mathematics, a Fourier series is a way to represent a function as the sum of simple sine waves. More formally, it decomposes any periodic function or periodic signal into the sum of a set of simple oscillating functions, the discrete-time Fourier transform is a periodic function, often defined in terms of a Fourier series. The Z-transform, another example of application, reduces to a Fourier series for the important case |z|=1, Fourier series are also central to the original proof of the Nyquist–Shannon sampling theorem. The study of Fourier series is a branch of Fourier analysis, the Mémoire introduced Fourier analysis, specifically Fourier series. Through Fouriers research the fact was established that a function can be represented by a trigonometric series. The first announcement of this discovery was made by Fourier in 1807. The heat equation is a differential equation. These simple solutions are now sometimes called eigensolutions, Fouriers idea was to model a complicated heat source as a superposition of simple sine and cosine waves, and to write the solution as a superposition of the corresponding eigensolutions. This superposition or linear combination is called the Fourier series, from a modern point of view, Fouriers results are somewhat informal, due to the lack of a precise notion of function and integral in the early nineteenth century. Later, Peter Gustav Lejeune Dirichlet and Bernhard Riemann expressed Fouriers results with greater precision, in this section, s denotes a function of the real variable x, and s is integrable on an interval, for real numbers x0 and P. We will attempt to represent s in that interval as a sum, or series. Outside the interval, the series is periodic with period P and it follows that if s also has that property, the approximation is valid on the entire real line. We can begin with a summation, s N = A02 + ∑ n =1 N A n ⋅ sin . S N is a function with period P. The inverse relationships between the coefficients are, A n = a n 2 + b n 2 ϕ n = atan2 , when the coefficients are computed as follows, s N approximates s on, and the approximation improves as N → ∞. The infinite sum, s ∞, is called the Fourier series representation of s, both components of a complex-valued function are real-valued functions that can be represented by a Fourier series. This is the formula as before except cn and c−n are no longer complex conjugates. In particular, the Fourier series converges absolutely and uniformly to s whenever the derivative of s is square integrable, if a function is square-integrable on the interval, then the Fourier series converges to the function at almost every point