1.
Circle
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A circle is a simple closed shape in Euclidean geometry. The distance between any of the points and the centre is called the radius, a circle is a simple closed curve which divides the plane into two regions, an interior and an exterior. Annulus, the object, the region bounded by two concentric circles. Arc, any connected part of the circle, centre, the point equidistant from the points on the circle. Chord, a segment whose endpoints lie on the circle. Circumference, the length of one circuit along the circle, or the distance around the circle and it is a special case of a chord, namely the longest chord, and it is twice the radius. Disc, the region of the bounded by a circle. Lens, the intersection of two discs, passant, a coplanar straight line that does not touch the circle. Radius, a line segment joining the centre of the circle to any point on the circle itself, or the length of such a segment, sector, a region bounded by two radii and an arc lying between the radii. Segment, a region, not containing the centre, bounded by a chord, secant, an extended chord, a coplanar straight line cutting the circle at two points. Semicircle, an arc that extends from one of a diameters endpoints to the other, in non-technical common usage it may mean the diameter, arc, and its interior, a two dimensional region, that is technically called a half-disc. A half-disc is a case of a segment, namely the largest one. Tangent, a straight line that touches the circle at a single point. The word circle derives from the Greek κίρκος/κύκλος, itself a metathesis of the Homeric Greek κρίκος, the origins of the words circus and circuit are closely related. The circle has been known since before the beginning of recorded history, natural circles would have been observed, such as the Moon, Sun, and a short plant stalk blowing in the wind on sand, which forms a circle shape in the sand. The circle is the basis for the wheel, which, with related inventions such as gears, in mathematics, the study of the circle has helped inspire the development of geometry, astronomy and calculus. Some highlights in the history of the circle are,1700 BCE – The Rhind papyrus gives a method to find the area of a circular field. The result corresponds to 256/81 as a value of π.300 BCE – Book 3 of Euclids Elements deals with the properties of circles

2.
Parametric equation
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In mathematics, parametric equations define a group of quantities as functions of one or more independent variables called parameters. For example, the equations x = cos t y = sin t form a representation of the unit circle. Parametric equations are used in kinematics, where the trajectory of an object is represented by equations depending on time as the parameter. Because of this application, a parameter is often labeled t, however. Parameterizations are non-unique, more than one set of equations can specify the same curve. In kinematics, objects paths through space are described as parametric curves. Used in this way, the set of equations for the objects coordinates collectively constitute a vector-valued function for position. Such parametric curves can then be integrated and differentiated termwise, thus, if a particles position is described parametrically as r = then its velocity can be found as v = r ′ = and its acceleration as a = r ″ =. Another important use of equations is in the field of computer-aided design. For example, consider the three representations, all of which are commonly used to describe planar curves. These problems can be addressed by rewriting the non-parametric equations in parametric form, numerous problems in integer geometry can be solved using parametric equations. A classical such solution is Euclids parametrization of right triangles such that the lengths of their sides a, b, by multiplying a, b and c by an arbitrary positive integer, one gets a parametrization of all right triangles whose three sides have integer lengths. Converting a set of equations to a single equation involves eliminating the variable t from the simultaneous equations x = x, y = y. If one of these equations can be solved for t, the expression obtained can be substituted into the equation to obtain an equation involving x and y only. If the parametrization is given by rational functions x = p r, y = q r, where p, q, r are set-wise coprime polynomials, in some cases there is no single equation in closed form that is equivalent to the parametric equations. The simplest equation for a parabola, y = x 2 can be parameterized by using a free parameter t, and setting x = t, y = t 2 f o r − ∞ < t < ∞. More generally, any given by an explicit equation y = f can be parameterized by using a free parameter t. A more sophisticated example is the following, consider the unit circle which is described by the ordinary equation x 2 + y 2 =1

3.
Geocentric model
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In astronomy, the geocentric model is a superseded description of the universe with the Earth at the center. Under the geocentric model, the Sun, Moon, stars, the geocentric model served as the predominant description of the cosmos in many ancient civilizations, such as those of Aristotle and Ptolemy. Two observations supported the idea that the Earth was the center of the Universe, first, the Sun appears to revolve around the Earth once per day. While the Moon and the planets have their own motions, they appear to revolve around the Earth about once per day. The stars appeared to be on a sphere, rotating once each day along an axis through the north and south geographic poles of the Earth. Second, the Earth does not seem to move from the perspective of an Earth-bound observer, it appears to be solid, stable, Ancient Greek, ancient Roman and medieval philosophers usually combined the geocentric model with a spherical Earth. It is not the same as the older flat Earth model implied in some mythology, the ancient Jewish Babylonian uranography pictured a flat Earth with a dome-shaped rigid canopy named firmament placed over it. The astronomical predictions of Ptolemys geocentric model were used to prepare astrological and astronomical charts for over 1500 years. The geocentric model held sway into the modern age, but from the late 16th century onward, it was gradually superseded by the Heliocentric model of Copernicus, Galileo. There was much resistance to the transition between these two theories, christian theologians were reluctant to reject a theory that agreed with Bible passages. Others felt a new, unknown theory could not subvert an accepted consensus for geocentrism, the geocentric model entered Greek astronomy and philosophy at an early point, it can be found in Pre-Socratic philosophy. In the 6th century BC, Anaximander proposed a cosmology with the Earth shaped like a section of a pillar, the Sun, Moon, and planets were holes in invisible wheels surrounding the Earth, through the holes, humans could see concealed fire. About the same time, the Pythagoreans thought that the Earth was a sphere, later these views were combined, so most educated Greeks from the 4th century BC on thought that the Earth was a sphere at the center of the universe. In the 4th century BC, two influential Greek philosophers, Plato and his student Aristotle, wrote works based on the geocentric model, according to Plato, the Earth was a sphere, stationary at the center of the universe. In his Myth of Er, a section of the Republic, Plato describes the cosmos as the Spindle of Necessity, attended by the Sirens and these spheres, known as crystalline spheres, all moved at different uniform speeds to create the revolution of bodies around the Earth. They were composed of a substance called aether. Aristotle believed that the moon was in the innermost sphere and therefore touches the realm of Earth, causing the dark spots and he further described his system by explaining the natural tendencies of the terrestrial elements, Earth, water, fire, air, as well as celestial aether. His system held that Earth was the heaviest element, with the strongest movement towards the center, the tendency of air and fire, on the other hand, was to move upwards, away from the center, with fire being lighter than air

4.
Wankel engine
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The Wankel engine is a type of internal combustion engine using an eccentric rotary design to convert pressure into rotating motion. The engine is referred to as a rotary engine, although this name applies also to other completely different designs. All parts rotate moving in one direction, as opposed to the reciprocating piston engine which has pistons violently changing direction. The design was conceived by German engineer Felix Wankel, Wankel received his first patent for the engine in 1929. He began development in the early 1950s at NSU, and completed a prototype in 1957. NSU subsequently licensed the design to companies around the world, who have continually added improvements, the engines produced are of spark ignition, with compression ignition engines only in research projects. The Wankel engine has the advantages of compact design and low weight over the most commonly used internal combustion engine employing reciprocating pistons, the point of power to weight has been reached of under one pound weight per horsepower output. In 1951, NSU Motorenwerke AG in Germany began development of the engine, the first, the DKM motor, was developed by Felix Wankel. The second, the KKM motor, developed by Hanns Dieter Paschke, was adopted as the basis of the modern Wankel engine, so the Wankel engine design used today was not designed by Felix Wankel, and the Paschke engine could be a more apt title. The basis of the DKM type of motor was that both the rotor and the housing spun around on separate axes, the DKM motor reached higher revolutions per minute and was more naturally balanced. However, the engine needed to be stripped to change the spark plugs, the KKM engine was simpler, having a fixed housing. The first working prototype, DKM54, produced 21 hp and ran on February 1,1957, at the NSU research and development department Versuchsabteilung TX. The KKM57 was constructed by NSU engineer Hanns Dieter Paschke in 1957 without the knowledge of Felix Wankel, in 1960, NSU, the firm that employed the two inventors, and the US firm Curtiss-Wright, signed a joint agreement. Curtiss-Wright recruited Max Bentele to head their design team, many manufacturers signed license agreements for development, attracted by the smoothness, quiet running, and reliability emanating from the uncomplicated design. Amongst them were Alfa Romeo, American Motors, Citroen, Ford, General Motors, Mazda, Mercedes-Benz, Nissan, Porsche, Rolls-Royce, Suzuki, in the United States in 1959, under license from NSU, Curtiss-Wright pioneered improvements in the basic engine design. In Britain, in the 1960s, Rolls Royces Motor Car Division pioneered a two-stage diesel version of the Wankel engine, Citroën did much research, producing the M35, GS Birotor and RE-2 Helicopter using engines produced by Comotor, a joint venture of Citroën and NSU. General Motors seemed to have concluded the Wankel engine was more expensive to build than an equivalent reciprocating engine. General Motors claimed to have solved the fuel issue

5.
University of St Andrews
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The University of St Andrews is a British public research university in St Andrews, Fife, Scotland. It is the oldest of the four ancient universities of Scotland, St Andrews was founded between 1410 and 1413, when the Avignon Antipope Benedict XIII issued a papal bull to a small founding group of Augustinian clergy. St Andrews is made up from a variety of institutions, including three constituent colleges and 18 academic schools organised into four faculties, the university occupies historic and modern buildings located throughout the town. The academic year is divided into two terms, Martinmas and Candlemas, in term time, over one-third of the towns population is either a staff member or student of the university. It is ranked as the third best university in the United Kingdom in national league tables, the Times Higher Education World Universities Ranking names St Andrews among the worlds Top 50 universities for Social Sciences, Arts and Humanities. St Andrews has the highest student satisfaction amongst all multi-faculty universities in the United Kingdom, St Andrews has many notable alumni and affiliated faculty, including eminent mathematicians, scientists, theologians, philosophers, and politicians. Six Nobel Laureates are among St Andrews alumni and former staff, a charter of privilege was bestowed upon the society of masters and scholars by the Bishop of St Andrews, Henry Wardlaw, on 28 February 1411. Wardlaw then successfully petitioned the Avignon Pope Benedict XIII to grant the university status by issuing a series of papal bulls. King James I of Scotland confirmed the charter of the university in 1432, subsequent kings supported the university with King James V confirming privileges of the university in 1532. A college of theology and arts called St Johns College was founded in 1418 by Robert of Montrose, St Salvators College was established in 1450, by Bishop James Kennedy. St Leonards College was founded in 1511 by Archbishop Alexander Stewart, St Johns College was refounded by Cardinal James Beaton under the name St Marys College in 1538 for the study of divinity and law. Some university buildings that date from this period are still in use today, such as St Salvators Chapel, St Leonards College Chapel, at this time, the majority of the teaching was of a religious nature and was conducted by clerics associated with the cathedral. During the 17th and 18th centuries, the university had mixed fortunes and was beset by civil. He described it as pining in decay and struggling for life, in the second half of the 19th century, pressure was building upon universities to open up higher education to women. In 1876, the University Senate decided to allow women to receive an education at St Andrews at a roughly equal to the Master of Arts degree that men were able to take at the time. The scheme came to be known as the L. L. A and it required women to pass five subjects at an ordinary level and one at honours level and entitled them to hold a degree from the university. In 1889 the Universities Act made it possible to admit women to St Andrews. Agnes Forbes Blackadder became the first woman to graduate from St Andrews on the level as men in October 1894

6.
Combustion chamber
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A combustion chamber is that part of an internal combustion engine in which the fuel/air mix is burned. ICEs typically comprise reciprocating piston engines, rotary engines, gas turbines, the combustion process increases the internal energy of a gas, which translates into an increase in temperature, pressure, or volume depending on the configuration. In an enclosure, for example the cylinder of an engine, the volume is controlled. In a continuous system, for example a jet engine combustor, the pressure is controlled. This increase in pressure or volume can be used to do work, for example, if the gas velocity changes, thrust is produced, such as in the nozzle of a rocket engine. At top dead centre the pistons of an engine are flush with the top of the cylinder block. The combustion chamber may be a recess either in the cylinder head, a design with the combustion chamber in the piston is called a Heron head, where the head is machined flat but the pistons are dished. The Heron head has proved even more efficient than the hemispherical head. Intake valves permit the inflow of a fuel air mix, and this is best achieved with a compact rather than elongated chamber. Swirl & Squish The intake valve/port is usually placed to give the mixture a pronounced swirl above the piston, improving mixing. The shape of the top also affects the amount of swirl. Another design feature to promote turbulence for good fuel/air mixing is squish, where swirl is particularly important, combustion chambers in the piston may be favoured. Flame front Ignition typically occurs around 15 degrees before top dead centre, the spark plug must be sited so that the flame front can progress throughout the combustion chamber. Harry Ricardo was prominent in developing combustion chambers for diesel engines, the combustion chamber in gas turbines and jet engines is called the combustor. Different types of combustors exist, mainly, Can type, Can combustors are self-contained cylindrical combustion chambers, each can has its own fuel injector, liner, interconnectors, casing. Each can get an air source from individual opening, cannular type, Like the can type combustor, can annular combustors have discrete combustion zones contained in separate liners with their own fuel injectors. Unlike the can combustor, all the combustion zones share a common air casing, Annular type, Annular combustors do away with the separate combustion zones and simply have a continuous liner and casing in a ring. The term combustion chamber is used to refer to an additional space between the firebox and boiler in a steam locomotive

7.
Cycloid
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A cycloid is the curve traced by a point on the rim of a circular wheel as the wheel rolls along a straight line without slippage. It is an example of a roulette, a curve generated by a curve rolling on another curve, the cycloid has been called The Helen of Geometers as it caused frequent quarrels among 17th-century mathematicians. Historians of mathematics have proposed several candidates for the discoverer of the cycloid, mathematical historian Paul Tannery cited similar work by the Syrian philosopher Iamblichus as evidence that the curve was likely known in antiquity. Galileo Galileis name was put forward at the end of the 19th century, in this work, Bovelles mistakes the arch traced by a rolling wheel as part of a larger circle with a radius 120% larger than the smaller wheel. Galileo originated the term cycloid and was the first to make a study of the curve. He discovered the ratio was roughly 3,1 but incorrectly concluded the ratio was an irrational fraction, around 1628, Gilles Persone de Roberval likely learned of the quadrature problem from Père Marin Mersenne and effected the quadrature in 1634 by using Cavalieris Theorem. However, this work was not published until 1693, constructing the tangent of the cycloid dates to August 1638 when Mersenne received unique methods from Roberval, Pierre de Fermat and René Descartes. Mersenne passed these results along to Galileo, who gave them to his students Torricelli and Viviana and this result and others were published by Torricelli in 1644, which is also the first printed work on the cycloid. This led to Roberval charging Torricelli with plagiarism, with the cut short by Torricellis early death in 1647. In 1658, Blaise Pascal had given up mathematics for theology but, while suffering from a toothache and his toothache disappeared, and he took this as a heavenly sign to proceed with his research. Eight days later he had completed his essay and, to publicize the results, Pascal proposed three questions relating to the center of gravity, area and volume of the cycloid, with the winner or winners to receive prizes of 20 and 40 Spanish doubloons. Pascal, Roberval and Senator Carcavy were the judges, and neither of the two submissions were judged to be adequate. While the contest was ongoing, Christopher Wren sent Pascal a proposal for a proof of the rectification of the cycloid, wallis published Wrens proof in Walliss Tractus Duo, giving Wren priority for the first published proof. In 1686, Gottfried Wilhelm Leibniz used analytic geometry to describe the curve with a single equation, in 1696, Johann Bernoulli posed the brachistochrone problem, the solution of which is a cycloid. For given t, the centre lies at x = rt. Solving for t and replacing, the Cartesian equation is found to be, an equation for the cycloid of the form y = f with a closed-form expression for the right-hand side is not possible. When y is viewed as a function of x, the cycloid is differentiable everywhere except at the cusps, the map from t to is a differentiable curve or parametric curve of class C∞ and the singularity where the derivative is 0 is an ordinary cusp. A cycloid segment from one cusp to the next is called an arch of the cycloid, the first arch of the cycloid consists of points such that 0 ≤ t ≤2 π

8.
Spirograph
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Spirograph is a geometric drawing toy that produces mathematical roulette curves of the variety technically known as hypotrochoids and epitrochoids. It was developed by British engineer Denys Fisher and first sold in 1965, the name has been a registered trademark of Hasbro Inc. since 1998 following purchase of the company that had acquired the Denys Fisher company. The Spirograph brand was relaunched with original product configurations in the USA in 2013 by Kahootz Toys and in Europe by Goldfish, the mathematician Bruno Abakanowicz invented the Spirograph between 1881 and 1900. It was used for calculating an area delimited by curves, drawing toys based on gears have been around since at least 1908, when The Marvelous Wondergraph was advertised in the Sears catalog. An article describing how to make a Wondergraph drawing machine appeared in the Boys Mechanic publication in 1913, the Spirograph itself was developed by the British engineer Denys Fisher, who exhibited at the 1965 Nuremberg International Toy Fair. It was subsequently produced by his company, US distribution rights were acquired by Kenner, Inc. which introduced it to the United States market in 1966 and promoted it as a creative childrens toy. In 2013 the Spirograph brand was re-launched in the USA by Kahootz Toys and in Europe by Goldfish and Bison with products that returned to the use of the original gears and wheels. The modern products use removable putty in place of pins or are held down by hand to keep the pieces in place on the paper. The Spirograph was a 2014 Toy of the Year finalist in 2 categories, the original US-released Spirograph consisted of two different-sized plastic rings, with gear teeth on both the inside and outside of their circumferences. They were pinned to a backing with pins, and any of several provided gearwheels. It could be spun around to make geometric shapes on the paper medium. Later, the Super-Spirograph consisted of a set of gears and other interlocking shape-segments such as rings, triangles. It has several sizes of gears and shapes, and all edges have teeth to engage any other piece. For instance, smaller gears fit inside the rings. Kenner also introduced Spirotot, Magnetic Spirograph, Spiroman and various refill sets, to use it, a sheet of paper is placed on a heavy cardboard backing, and one of the plastic pieces—known as a stator—is secured via pins or reusable adhesive to the paper and cardboard. Another plastic piece—called the rotor—is placed so that its teeth engage with those of the pinned piece, for example, a ring may be pinned to the paper and a small gear placed inside the ring. The number of arrangements possible by combining different gears is very large, the point of a pen is placed in one of the holes of the rotor. As the rotor is moved, the pen traces out a curve, more intricate and unusual-shaped patterns may be made through the use of both hands, one to draw and one to guide the pieces

9.
Epicycloid
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In geometry, an epicycloid is a plane curve produced by tracing the path of a chosen point on the circumference of a circle — called an epicycle — which rolls without slipping around a fixed circle. It is a kind of roulette. If k is an integer, then the curve is closed, if k is a rational number, say k=p/q expressed in simplest terms, then the curve has p cusps. If k is a number, then the curve never closes. Epicycloid examples The epicycloid is a kind of epitrochoid. An epicycle with one cusp is a cardioid, two cusps is a nephroid, an epicycloid and its evolute are similar. We assume that the position of p is what we want to solve, α is the radian from the point to the moving point p. A catalog of special plane curves, Epicycloid by Michael Ford, The Wolfram Demonstrations Project,2007 OConnor, John J. Robertson, Edmund F. Epicycloid, MacTutor History of Mathematics archive, University of St Andrews. Animation of Epicycloids, Pericycloids and Hypocycloids Spirograph -- GeoFun Historical note on the application of the epicycloid to the form of Gear Teeth

10.
Hypocycloid
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In geometry, a hypocycloid is a special plane curve generated by the trace of a fixed point on a small circle that rolls within a larger circle. It is comparable to the cycloid but instead of the circle rolling along a line, if k is an integer, then the curve is closed, and has k cusps. Specially for k=2 the curve is a line and the circles are called Cardano circles. Girolamo Cardano was the first to describe these hypocycloids and their applications to high-speed printing, if k is a rational number, say k = p/q expressed in simplest terms, then the curve has p cusps. If k is a number, then the curve never closes. A hypocycloid with three cusps is known as a deltoid, a hypocycloid curve with four cusps is known as an astroid. The hypocycloid with two cusps is a degenerate but still very interesting case, known as the Tusi couple and this motion looks like rolling, though it is not technically rolling in the sense of classical mechanics, since it involves slipping. Hypocycloid shapes can be related to special groups, denoted SU. For example, the values of the sum of diagonal entries for a matrix in SU, are precisely the points in the complex plane lying inside a hypocycloid of three cusps. Likewise, summing the entries of SU matrices give points inside an astroid. Thanks to this result, one can use the fact that SU fits inside SU as a subgroup to prove that an epicycloid with k cusps moves snugly inside one with k+1 cusps. The evolute of a hypocycloid is a version of the hypocycloid itself. The pedal of a hypocycloid with pole at the center of the hypocycloid is a rose curve, the isoptic of a hypocycloid is a hypocycloid. Curves similar to hypocyloids can be drawn with the Spirograph toy, specifically, the Spirograph can draw hypotrochoids and epitrochoids. The Pittsburgh Steelers logo, which is based on the Steelmark, in his weekly NFL. com column Tuesday Morning Quarterback, Gregg Easterbrook often refers to the Steelers as the Hypocycloids. Chilean soccer team CD Huachipato based their crest on the Steelers logo, the first Drew Carey season of The Price Is Rights set features astroids on the three main doors, giant price tag, and the turntable area. The astroids on the doors and turntable were removed when the show switched to high definition starting in 2008. Special cases, Astroid, Deltoid List of periodic functions Cyclogon Epicycloid Hypotrochoid Epitrochoid Spirograph Flag of Portland, Oregon, a catalog of special plane curves