1.
Plane (geometry)
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In mathematics, a plane is a flat, two-dimensional surface that extends infinitely far. A plane is the analogue of a point, a line. When working exclusively in two-dimensional Euclidean space, the article is used, so. Many fundamental tasks in mathematics, geometry, trigonometry, graph theory and graphing are performed in a space, or in other words. Euclid set forth the first great landmark of mathematical thought, a treatment of geometry. He selected a small core of undefined terms and postulates which he used to prove various geometrical statements. Although the plane in its sense is not directly given a definition anywhere in the Elements. In his work Euclid never makes use of numbers to measure length, angle, in this way the Euclidean plane is not quite the same as the Cartesian plane. This section is concerned with planes embedded in three dimensions, specifically, in R3. In a Euclidean space of any number of dimensions, a plane is determined by any of the following. A line and a point not on that line, a line is either parallel to a plane, intersects it at a single point, or is contained in the plane. Two distinct lines perpendicular to the plane must be parallel to each other. Two distinct planes perpendicular to the line must be parallel to each other. Specifically, let r0 be the vector of some point P0 =. The plane determined by the point P0 and the vector n consists of those points P, with position vector r, such that the vector drawn from P0 to P is perpendicular to n. Recalling that two vectors are perpendicular if and only if their dot product is zero, it follows that the plane can be described as the set of all points r such that n ⋅ =0. Expanded this becomes a + b + c =0, which is the form of the equation of a plane. This is just a linear equation a x + b y + c z + d =0 and this familiar equation for a plane is called the general form of the equation of the plane
2.
Mathematics
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Mathematics is the study of topics such as quantity, structure, space, and change. There is a range of views among mathematicians and philosophers as to the exact scope, Mathematicians seek out patterns and use them to formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proof, when mathematical structures are good models of real phenomena, then mathematical reasoning can provide insight or predictions about nature. Through the use of abstraction and logic, mathematics developed from counting, calculation, measurement, practical mathematics has been a human activity from as far back as written records exist. The research required to solve mathematical problems can take years or even centuries of sustained inquiry, rigorous arguments first appeared in Greek mathematics, most notably in Euclids Elements. Galileo Galilei said, The universe cannot be read until we have learned the language and it is written in mathematical language, and the letters are triangles, circles and other geometrical figures, without which means it is humanly impossible to comprehend a single word. Without these, one is wandering about in a dark labyrinth, carl Friedrich Gauss referred to mathematics as the Queen of the Sciences. Benjamin Peirce called mathematics the science that draws necessary conclusions, David Hilbert said of mathematics, We are not speaking here of arbitrariness in any sense. Mathematics is not like a game whose tasks are determined by arbitrarily stipulated rules, rather, it is a conceptual system possessing internal necessity that can only be so and by no means otherwise. Albert Einstein stated that as far as the laws of mathematics refer to reality, they are not certain, Mathematics is essential in many fields, including natural science, engineering, medicine, finance and the social sciences. Applied mathematics has led to entirely new mathematical disciplines, such as statistics, Mathematicians also engage in pure mathematics, or mathematics for its own sake, without having any application in mind. There is no clear line separating pure and applied mathematics, the history of mathematics can be seen as an ever-increasing series of abstractions. The earliest uses of mathematics were in trading, land measurement, painting and weaving patterns, in Babylonian mathematics elementary arithmetic first appears in the archaeological record. Numeracy pre-dated writing and numeral systems have many and diverse. Between 600 and 300 BC the Ancient Greeks began a study of mathematics in its own right with Greek mathematics. Mathematics has since been extended, and there has been a fruitful interaction between mathematics and science, to the benefit of both. Mathematical discoveries continue to be made today, the overwhelming majority of works in this ocean contain new mathematical theorems and their proofs. The word máthēma is derived from μανθάνω, while the modern Greek equivalent is μαθαίνω, in Greece, the word for mathematics came to have the narrower and more technical meaning mathematical study even in Classical times
3.
Smoothness
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In mathematical analysis, the smoothness of a function is a property measured by the number of derivatives it has which are continuous. A smooth function is a function that has derivatives of all orders everywhere in its domain, differentiability class is a classification of functions according to the properties of their derivatives. Higher order differentiability classes correspond to the existence of more derivatives, consider an open set on the real line and a function f defined on that set with real values. Let k be a non-negative integer, the function f is said to be of class Ck if the derivatives f′, f′′. The function f is said to be of class C∞, or smooth, if it has derivatives of all orders. The function f is said to be of class Cω, or analytic, if f is smooth, Cω is thus strictly contained in C∞. Bump functions are examples of functions in C∞ but not in Cω, to put it differently, the class C0 consists of all continuous functions. The class C1 consists of all differentiable functions whose derivative is continuous, thus, a C1 function is exactly a function whose derivative exists and is of class C0. In particular, Ck is contained in Ck−1 for every k, C∞, the class of infinitely differentiable functions, is the intersection of the sets Ck as k varies over the non-negative integers. The function f = { x if x ≥0,0 if x <0 is continuous, because cos oscillates as x →0, f ’ is not continuous at zero. Therefore, this function is differentiable but not of class C1, the functions f = | x | k +1 where k is even, are continuous and k times differentiable at all x. But at x =0 they are not times differentiable, so they are of class Ck, the exponential function is analytic, so, of class Cω. The trigonometric functions are also analytic wherever they are defined, the function f is an example of a smooth function with compact support. Let n and m be some positive integers, if f is a function from an open subset of Rn with values in Rm, then f has component functions f1. Each of these may or may not have partial derivatives, the classes C∞ and Cω are defined as before. These criteria of differentiability can be applied to the functions of a differential structure. The resulting space is called a Ck manifold, if one wishes to start with a coordinate-independent definition of the class Ck, one may start by considering maps between Banach spaces. A map from one Banach space to another is differentiable at a point if there is a map which approximates it at that point
4.
Plane curve
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In mathematics, a plane curve is a curve in a plane that may be either a Euclidean plane, an affine plane or a projective plane. The most frequently studied cases are smooth plane curves, and algebraic plane curves, a smooth plane curve is a curve in a real Euclidean plane R2 and is a one-dimensional smooth manifold. This means that a plane curve is a plane curve which locally looks like a line, in the sense that near every point. An algebraic plane curve is a curve in an affine or projective plane given by one polynomial equation f =0 Algebraic curves were studied extensively since the 18th century, for example, the circle given by the equation x2 + y2 =1 has degree 2. The non-singular plane algebraic curves of degree 2 are called conic sections, the plane curves of degree 3 are called cubic plane curves and, if they are non-singular, elliptic curves. Those of degree four are called plane curves. Algebraic curve Differential geometry Algebraic geometry Plane curve fitting Projective varieties Two-dimensional graph Coolidge, a Treatise on Algebraic Plane Curves, Dover Publications, ISBN 0-486-49576-0. A handbook on curves and their properties, J. W. Edwards, ASIN B0007EKXV0
5.
Connected component (graph theory)
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For example, the graph shown in the illustration on the right has three connected components. A vertex with no incident edges is itself a connected component, a graph that is itself connected has exactly one connected component, consisting of the whole graph. An alternative way to define connected components involves the equivalence classes of a relation that is defined on the vertices of the graph. In an undirected graph, a v is reachable from a vertex u if there is a path from u to v. In this definition, a vertex is counted as a path of length zero. Reachability is a relation, since, It is reflexive. It is symmetric, If there is a path from u to v and it is transitive, If there is a path from u to v and a path from v to w, the two paths may be concatenated together to form a path from u to w. The connected components are then the induced subgraphs formed by the classes of this relation. The number of connected components is an important topological invariant of a graph, in topological graph theory it can be interpreted as the zeroth Betti number of the graph. In algebraic graph theory it equals the multiplicity of 0 as an eigenvalue of the Laplacian matrix of the graph and it is also the index of the first nonzero coefficient of the chromatic polynomial of a graph. Numbers of connected components play a key role in the Tutte theorem characterizing graphs that have perfect matchings and it is straightforward to compute the connected components of a graph in linear time using either breadth-first search or depth-first search. In either case, a search that begins at some vertex v will find the entire connected component containing v before returning. Hopcroft and Tarjan describe essentially this algorithm, and state that at that point it was well known, There are also efficient algorithms to dynamically track the connected components of a graph as vertices and edges are added, as a straightforward application of disjoint-set data structures. For forests, the cost can be reduced to O, or O amortized cost per edge deletion, finally Reingold succeeded in finding an algorithm for solving this connectivity problem in logarithmic space, showing that L = SL. Papadimitriou, Christos H. Symmetric space-bounded computation, Theoretical Computer Science,19, 161–187, Reingold, Omer, Undirected connectivity in log-space, Journal of the ACM,55, Article 17,24 pages, doi,10. 1145/1391289.1391291. MATLAB code to find connected components in undirected graphs, MATLAB File Exchange, connected components, Steven Skiena, The Stony Brook Algorithm Repository
6.
Bow and arrow
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The bow and arrow is a projectile weapon system that predates recorded history and is common to most cultures. Archery is the art, practice, or skill of applying it, a bow is a flexible arc that shoots aerodynamic projectiles called arrows. A string joins the two ends of the bow and when the string is drawn back, the ends of the bow are flexed, when the string is released, the potential energy of the flexed stick is transformed into the kinetic energy of the arrow. Archery is the art or sport of shooting arrows from bows, today, bows and arrows are used primarily for hunting and for the sport of archery. Someone who makes bows is known as a bowyer, and one who makes arrows is a fletcher —or in the case of the manufacture of arrow heads. The bow and arrow appears around the transition from the Upper Paleolithic to the Mesolithic, at the site of Nataruk in Turkana County, Kenya, obsidian bladelets found embedded in a skull and within the thoracic cavity of another skeleton, suggest the use of stone-tipped arrows as weapons. After the end of the last glacial period, use of the bow seems to have spread to every inhabited continent, including the New World, the oldest extant bows in one piece are the elm Holmegaard bows from Denmark which were dated to 9,000 BCE. High-performance wooden bows are made following the Holmegaard design. Microliths discovered on the south coast of Africa suggest that arrows may be at least 71,000 years old, the bow was an important weapon for both hunting and warfare from prehistoric times until the widespread use of gunpowder in the 16th century. Organised warfare with bows ended in the mid 17th century in Europe, the British upper class led a revival of archery from the late 18th century. Sir Ashton Lever, an antiquarian and collector, formed the Toxophilite Society in London in 1781, under the patronage of George, the basic elements of a bow are a pair of curved elastic limbs, traditionally made from wood, joined by a riser. Both ends of the limbs are connected by a known as the bow string. By pulling the string backwards the archer exerts compressive force on the section, or belly, of the limbs as well as placing the outer section, or back. While the string is held, this stores the energy released in putting the arrow to flight. The force required to hold the string stationary at full draw is used to express the power of a bow. Other things being equal, a draw weight means a more powerful bow. The various parts of the bow can be subdivided into further sections, the topmost limb is known as the upper limb, while the bottom limb is the lower limb. At the tip of each limb is a nock, which is used to attach the bowstring to the limbs, the riser is usually divided into the grip, which is held by the archer, as well as the arrow rest and the bow window
7.
Conic section
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In mathematics, a conic section is a curve obtained as the intersection of the surface of a cone with a plane. The three types of conic section are the hyperbola, the parabola, and the ellipse, the circle is a special case of the ellipse, and is of sufficient interest in its own right that it was sometimes called a fourth type of conic section. The conic sections have been studied by the ancient Greek mathematicians with this work culminating around 200 BC, the conic sections of the Euclidean plane have various distinguishing properties. Many of these have used as the basis for a definition of the conic sections. The type of conic is determined by the value of the eccentricity, in analytic geometry, a conic may be defined as a plane algebraic curve of degree 2, that is, as the set of points whose coordinates satisfy a quadratic equation in two variables. This equation may be written in form, and some geometric properties can be studied as algebraic conditions. In the Euclidean plane, the conic sections appear to be different from one another. By extending the geometry to a projective plane this apparent difference vanishes, further extension, by expanding the real coordinates to admit complex coordinates, provides the means to see this unification algebraically. The conic sections have been studied for thousands of years and have provided a source of interesting. A conic is the curve obtained as the intersection of a plane, called the cutting plane and we shall assume that the cone is a right circular cone for the purpose of easy description, but this is not required, any double cone with some circular cross-section will suffice. Planes that pass through the vertex of the cone will intersect the cone in a point and these are called degenerate conics and some authors do not consider them to be conics at all. Unless otherwise stated, we assume that conic refers to a non-degenerate conic. There are three types of conics, the ellipse, parabola, and hyperbola, the circle is a special kind of ellipse, although historically it had been considered as a fourth type. The circle and the ellipse arise when the intersection of the cone and plane is a closed curve, if the cutting plane is parallel to exactly one generating line of the cone, then the conic is unbounded and is called a parabola. In the remaining case, the figure is a hyperbola, in this case, the plane will intersect both halves of the cone, producing two separate unbounded curves. A property that the conic sections share is often presented as the following definition, a conic section is the locus of all points P whose distance to a fixed point F is a constant multiple of the distance from P to a fixed line L. For 0 < e <1 we obtain an ellipse, for e =1 a parabola, a circle is a limiting case and is not defined by a focus and directrix, in the plane. The eccentricity of a circle is defined to be zero and its focus is the center of the circle, an ellipse and a hyperbola each have two foci and distinct directrices for each of them
8.
Cone
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A cone is a three-dimensional geometric shape that tapers smoothly from a flat base to a point called the apex or vertex. A cone is formed by a set of segments, half-lines, or lines connecting a common point. If the enclosed points are included in the base, the cone is a solid object, otherwise it is a two-dimensional object in three-dimensional space. In the case of an object, the boundary formed by these lines or partial lines is called the lateral surface, if the lateral surface is unbounded. In the case of line segments, the cone does not extend beyond the base, while in the case of half-lines, in the case of lines, the cone extends infinitely far in both directions from the apex, in which case it is sometimes called a double cone. Either half of a cone on one side of the apex is called a nappe. The axis of a cone is the line, passing through the apex. If the base is right circular the intersection of a plane with this surface is a conic section, in general, however, the base may be any shape and the apex may lie anywhere. Contrasted with right cones are oblique cones, in which the axis passes through the centre of the base non-perpendicularly, a cone with a polygonal base is called a pyramid. Depending on the context, cone may also mean specifically a convex cone or a projective cone, cones can also be generalized to higher dimensions. The perimeter of the base of a cone is called the directrix, the base radius of a circular cone is the radius of its base, often this is simply called the radius of the cone. The aperture of a circular cone is the maximum angle between two generatrix lines, if the generatrix makes an angle θ to the axis, the aperture is 2θ. A cone with a region including its apex cut off by a plane is called a cone, if the truncation plane is parallel to the cones base. An elliptical cone is a cone with an elliptical base, a generalized cone is the surface created by the set of lines passing through a vertex and every point on a boundary. The slant height of a circular cone is the distance from any point on the circle to the apex of the cone. It is given by r 2 + h 2, where r is the radius of the cirf the cone and this application is primarily useful in determining the slant height of a cone when given other information regarding the radius or height. The volume V of any conic solid is one third of the product of the area of the base A B and the height h V =13 A B h. In modern mathematics, this formula can easily be computed using calculus – it is, up to scaling, the integral ∫ x 2 d x =13 x 3
9.
Parabola
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A parabola is a two-dimensional, mirror-symmetrical curve, which is approximately U-shaped when oriented as shown in the diagram below, but which can be in any orientation in its plane. It fits any of several different mathematical descriptions which can all be proved to define curves of exactly the same shape. One description of a parabola involves a point and a line, the focus does not lie on the directrix. The parabola is the locus of points in that plane that are equidistant from both the directrix and the focus, a parabola is a graph of a quadratic function, y = x2, for example. The line perpendicular to the directrix and passing through the focus is called the axis of symmetry, the point on the parabola that intersects the axis of symmetry is called the vertex, and is the point where the parabola is most sharply curved. The distance between the vertex and the focus, measured along the axis of symmetry, is the focal length, the latus rectum is the chord of the parabola which is parallel to the directrix and passes through the focus. Parabolas can open up, down, left, right, or in some arbitrary direction. Any parabola can be repositioned and rescaled to fit exactly on any other parabola — that is, conversely, light that originates from a point source at the focus is reflected into a parallel beam, leaving the parabola parallel to the axis of symmetry. The same effects occur with sound and other forms of energy and this reflective property is the basis of many practical uses of parabolas. The parabola has many important applications, from an antenna or parabolic microphone to automobile headlight reflectors to the design of ballistic missiles. They are frequently used in physics, engineering, and many other areas, the earliest known work on conic sections was by Menaechmus in the fourth century BC. He discovered a way to solve the problem of doubling the cube using parabolas, the name parabola is due to Apollonius who discovered many properties of conic sections. It means application, referring to application of concept, that has a connection with this curve. The focus–directrix property of the parabola and other conics is due to Pappus, Galileo showed that the path of a projectile follows a parabola, a consequence of uniform acceleration due to gravity. The idea that a reflector could produce an image was already well known before the invention of the reflecting telescope. Designs were proposed in the early to mid seventeenth century by many mathematicians including René Descartes, Marin Mersenne, when Isaac Newton built the first reflecting telescope in 1668, he skipped using a parabolic mirror because of the difficulty of fabrication, opting for a spherical mirror. Parabolic mirrors are used in most modern reflecting telescopes and in satellite dishes, solving for y yields y =14 f x 2. The length of the chord through the focus is called latus rectum, one half of it semi latus rectum
10.
Ellipse
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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
11.
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
12.
Cartesian coordinate system
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Each reference line is called a coordinate axis or just axis of the system, and the point where they meet is its origin, usually at ordered pair. The coordinates can also be defined as the positions of the projections of the point onto the two axis, expressed as signed distances from the origin. One can use the principle to specify the position of any point in three-dimensional space by three Cartesian coordinates, its signed distances to three mutually perpendicular planes. In general, n Cartesian coordinates specify the point in an n-dimensional Euclidean space for any dimension n and these coordinates are equal, up to sign, to distances from the point to n mutually perpendicular hyperplanes. The invention of Cartesian coordinates in the 17th century by René Descartes revolutionized mathematics by providing the first systematic link between Euclidean geometry and algebra. Using the Cartesian coordinate system, geometric shapes can be described by Cartesian equations, algebraic equations involving the coordinates of the points lying on the shape. For example, a circle of radius 2, centered at the origin of the plane, a familiar example is the concept of the graph of a function. Cartesian coordinates are also tools for most applied disciplines that deal with geometry, including astronomy, physics, engineering. They are the most common system used in computer graphics, computer-aided geometric design. Nicole Oresme, a French cleric and friend of the Dauphin of the 14th Century, used similar to Cartesian coordinates well before the time of Descartes. The adjective Cartesian refers to the French mathematician and philosopher René Descartes who published this idea in 1637 and it was independently discovered by Pierre de Fermat, who also worked in three dimensions, although Fermat did not publish the discovery. Both authors used a single axis in their treatments and have a length measured in reference to this axis. The concept of using a pair of axes was introduced later, after Descartes La Géométrie was translated into Latin in 1649 by Frans van Schooten and these commentators introduced several concepts while trying to clarify the ideas contained in Descartes work. Many other coordinate systems have developed since Descartes, such as the polar coordinates for the plane. The development of the Cartesian coordinate system would play a role in the development of the Calculus by Isaac Newton. The two-coordinate description of the plane was later generalized into the concept of vector spaces. Choosing a Cartesian coordinate system for a one-dimensional space – that is, for a straight line—involves choosing a point O of the line, a unit of length, and an orientation for the line. An orientation chooses which of the two half-lines determined by O is the positive, and which is negative, we say that the line is oriented from the negative half towards the positive half
13.
Sundial
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A sundial is a device that tells the time of day by the apparent position of the Sun in the sky. In the narrowest sense of the word it consists of a flat plate, as the sun appears to move across the sky, the shadow aligns with different hour-lines which are marked on the dial to indicate the time of day. The style is the edge of the gnomon, though a single point or nodus may be used. The gnomon casts a shadow, the shadow of the style shows the time. The gnomon may be a rod, a wire or a decorated metal casting. The style must be parallel to the axis of the Earths rotation for the sundial to be throughout the year. The styles angle from horizontal is equal to the geographical latitude. In a broader sense a sundial is any device that uses the suns altitude or azimuth to show the time, in addition to their time-telling function, sundials are valued as decorative objects, as literary metaphors and as objects of mathematical study. It is common for inexpensive mass-produced decorative sundials to have incorrectly aligned gnomons and hour-lines, there are several different types of sundials. Some sundials use a shadow or the edge of a shadow while others use a line or spot of light to indicate the time, the shadow-casting object, known as a gnomon, may be a long thin rod or other object with a sharp tip or a straight edge. Sundials employ many types of gnomon, the gnomon may be fixed or moved according to the season. It may be oriented vertically, horizontally, aligned with the Earths axis, given that sundials use light to indicate time, a line of light may be formed by allowing the suns rays through a thin slit or focusing them through a cylindrical lens. A spot of light may be formed by allowing the suns rays to pass through a hole or by reflecting them from a small circular mirror. Sundials also may use many types of surfaces to receive the light or shadow, planes are the most common surface, but partial spheres, cylinders, cones and other shapes have been used for greater accuracy or beauty. Sundials differ in their portability and their need for orientation, the installation of many dials requires knowing the local latitude, the precise vertical direction, and the direction to true North. Portable dials are self-aligning, for example, it may have two dials that operate on different principles, such as a horizontal and analemmatic dial, mounted together on one plate, in these designs, their times agree only when the plate is aligned properly. Sundials indicate the solar time, unless corrected for some other time. To obtain the clock time, three types of corrections need to be made
14.
Open orbit
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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
15.
Spacecraft
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A spacecraft is a vehicle, or machine designed to fly in outer space. Spacecraft are used for a variety of purposes, including communications, earth observation, meteorology, navigation, space colonization, planetary exploration, on a sub-orbital spaceflight, a spacecraft enters space and then returns to the surface, without having gone into an orbit. For orbital spaceflights, spacecraft enter closed orbits around the Earth or around other celestial bodies, robotic spacecraft used to support scientific research are space probes. Robotic spacecraft that remain in orbit around a body are artificial satellites. Only a handful of interstellar probes, such as Pioneer 10 and 11, Voyager 1 and 2, orbital spacecraft may be recoverable or not. By method of reentry to Earth they may be divided in non-winged space capsules, Sputnik 1 was the first artificial satellite. It was launched into an elliptical low Earth orbit by the Soviet Union on 4 October 1957, the launch ushered in new political, military, technological, and scientific developments, while the Sputnik launch was a single event, it marked the start of the Space Age. Apart from its value as a technological first, Sputnik 1 also helped to identify the upper atmospheric layers density and it also provided data on radio-signal distribution in the ionosphere. Pressurized nitrogen in the satellites false body provided the first opportunity for meteoroid detection, Sputnik 1 was launched during the International Geophysical Year from Site No. 1/5, at the 5th Tyuratam range, in Kazakh SSR. The satellite travelled at 29,000 kilometers per hour, taking 96.2 minutes to complete an orbit and this altitude is called the Kármán line. In particular, in the 1940s there were several test launches of the V-2 rocket, as of 2016, only three nations have flown manned spacecraft, USSR/Russia, USA, and China. The first manned spacecraft was Vostok 1, which carried Soviet cosmonaut Yuri Gagarin into space in 1961, there were five other manned missions which used a Vostok spacecraft. The second manned spacecraft was named Freedom 7, and it performed a sub-orbital spaceflight in 1961 carrying American astronaut Alan Shepard to an altitude of just over 187 kilometers, there were five other manned missions using Mercury spacecraft. Other Soviet manned spacecraft include the Voskhod, Soyuz, flown unmanned as Zond/L1, L3, TKS, China developed, but did not fly Shuguang, and is currently using Shenzhou. Except for the shuttle, all of the recoverable manned orbital spacecraft were space capsules. Manned space capsules The International Space Station, manned since November 2000, is a joint venture between Russia, the United States, Canada and several other countries, some reusable vehicles have been designed only for manned spaceflight, and these are often called spaceplanes. The first example of such was the North American X-15 spaceplane, the first reusable spacecraft, the X-15, was air-launched on a suborbital trajectory on July 19,1963. The first partially reusable spacecraft, a winged non-capsule, the Space Shuttle, was launched by the USA on the 20th anniversary of Yuri Gagarins flight
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Gravity assist
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Gravity assistance can be used to accelerate a spacecraft, that is, to increase or decrease its speed or redirect its path. The assist is provided by the motion of the body as it pulls on the spacecraft. It was used by interplanetary probes from Mariner 10 onwards, including the two Voyager probes notable flybys of Jupiter and Saturn, a gravity assist around a planet changes a spacecrafts velocity by entering and leaving the gravitational field of a planet. The spacecrafts speed increases as it approaches the planet and decreases while escaping its gravitational pull, because the planet orbits the sun, the spacecraft is affected by this motion during the maneuver. To increase speed, the flies with the movement of the planet, to decrease speed. The sum of the energies of both bodies remains constant. A slingshot maneuver can therefore be used to change the spaceships trajectory, a close terrestrial analogy is provided by a tennis ball bouncing off the front of a moving train. Imagine standing on a platform, and throwing a ball at 30 km/h toward a train approaching at 50 km/h. The driver of the sees the ball approaching at 80 km/h. Because of the motion, however, that departure is at 130 km/h relative to the train platform. Translating this analogy into space, in the reference frame. After the slingshot occurs and the leaves the planet, it will still have a velocity of v. In the Sun reference frame, the planet has a velocity of v, and by using the Pythagorean Theorem. After the spaceship leaves the planet, it will have a velocity of v + v = 2v and this example is also one of many trajectories and gained speeds the spaceship can have. The linear momentum gained by the spaceship is equal in magnitude to that lost by the planet, so the spacecraft gains velocity, however, the planets enormous mass compared to the spacecraft makes the resulting change in its speed negligibly small. These effects on the planet are so slight that they can be ignored in the calculation, realistic portrayals of encounters in space require the consideration of three dimensions. The same principles apply, only adding the planets velocity to that of the spacecraft requires vector addition, due to the reversibility of orbits, gravitational slingshots can also be used to reduce the speed of a spacecraft. Both Mariner 10 and MESSENGER performed this maneuver to reach Mercury, if even more speed is needed than available from gravity assist alone, the most economical way to utilize a rocket burn is to do it near the periapsis
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Escape velocity
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The escape velocity from Earth is about 11.186 km/s at the surface. More generally, escape velocity is the speed at which the sum of a kinetic energy. With escape velocity in a direction pointing away from the ground of a massive body, once escape velocity is achieved, no further impulse need be applied for it to continue in its escape. When given a speed V greater than the speed v e. In these equations atmospheric friction is not taken into account, escape velocity is only required to send a ballistic object on a trajectory that will allow the object to escape the gravity well of the mass M. The existence of escape velocity is a consequence of conservation of energy, by adding speed to the object it expands the possible places that can be reached until with enough energy they become infinite. For a given gravitational potential energy at a position, the escape velocity is the minimum speed an object without propulsion needs to be able to escape from the gravity. Escape velocity is actually a speed because it does not specify a direction, no matter what the direction of travel is, the simplest way of deriving the formula for escape velocity is to use conservation of energy. Imagine that a spaceship of mass m is at a distance r from the center of mass of the planet and its initial speed is equal to its escape velocity, v e. At its final state, it will be a distance away from the planet. The same result is obtained by a calculation, in which case the variable r represents the radial coordinate or reduced circumference of the Schwarzschild metric. All speeds and velocities measured with respect to the field, additionally, the escape velocity at a point in space is equal to the speed that an object would have if it started at rest from an infinite distance and was pulled by gravity to that point. In common usage, the point is on the surface of a planet or moon. On the surface of the Earth, the velocity is about 11.2 km/s. However, at 9,000 km altitude in space, it is less than 7.1 km/s. The escape velocity is independent of the mass of the escaping object and it does not matter if the mass is 1 kg or 1,000 kg, what differs is the amount of energy required. For an object of mass m the energy required to escape the Earths gravitational field is GMm / r, a related quantity is the specific orbital energy which is essentially the sum of the kinetic and potential energy divided by the mass. An object has reached escape velocity when the orbital energy is greater or equal to zero
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Comet
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A comet is an icy small Solar System body that, when passing close to the Sun, warms and begins to evolve gasses, a process called outgassing. This produces an atmosphere or coma, and sometimes also a tail. These phenomena are due to the effects of radiation and the solar wind acting upon the nucleus of the comet. Comet nuclei range from a few hundred metres to tens of kilometres across and are composed of collections of ice, dust. The coma may be up to 15 times the Earths diameter, if sufficiently bright, a comet may be seen from the Earth without the aid of a telescope and may subtend an arc of 30° across the sky. Comets have been observed and recorded since ancient times by many cultures, Comets usually have highly eccentric elliptical orbits, and they have a wide range of orbital periods, ranging from several years to potentially several millions of years. Short-period comets originate in the Kuiper belt or its associated scattered disc, long-period comets are thought to originate in the Oort cloud, a spherical cloud of icy bodies extending from outside the Kuiper belt to halfway to the nearest star. Long-period comets are set in motion towards the Sun from the Oort cloud by gravitational perturbations caused by passing stars, hyperbolic comets may pass once through the inner Solar System before being flung to interstellar space. The appearance of a comet is called an apparition, Comets are distinguished from asteroids by the presence of an extended, gravitationally unbound atmosphere surrounding their central nucleus. This atmosphere has parts termed the coma and the tail, however, extinct comets that have passed close to the Sun many times have lost nearly all of their volatile ices and dust and may come to resemble small asteroids. Asteroids are thought to have a different origin from comets, having formed inside the orbit of Jupiter rather than in the outer Solar System, the discovery of main-belt comets and active centaur minor planets has blurred the distinction between asteroids and comets. As of November 2014 there are 5,253 known comets, however, this represents only a tiny fraction of the total potential comet population, as the reservoir of comet-like bodies in the outer Solar System is estimated to be one trillion. Roughly one comet per year is visible to the eye, though many of those are faint. Particularly bright examples are called Great Comets, the word comet derives from the Old English cometa from the Latin comēta or comētēs. That, in turn, is a latinisation of the Greek κομήτης, Κομήτης was derived from κομᾶν, which was itself derived from κόμη and was used to mean the tail of a comet. The astronomical symbol for comets is ☄, consisting of a disc with three hairlike extensions. The solid, core structure of a comet is known as the nucleus, cometary nuclei are composed of an amalgamation of rock, dust, water ice, and frozen gases such as carbon dioxide, carbon monoxide, methane, and ammonia. As such, they are described as dirty snowballs after Fred Whipples model
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Subatomic particle
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In the physical sciences, subatomic particles are particles much smaller than atoms. There are two types of particles, elementary particles, which according to current theories are not made of other particles. Particle physics and nuclear physics study these particles and how they interact, in particle physics, the concept of a particle is one of several concepts inherited from classical physics. But it also reflects the understanding that at the quantum scale matter. The idea of a particle underwent serious rethinking when experiments showed that light could behave like a stream of particles as well as exhibit wave-like properties and this led to the new concept of wave–particle duality to reflect that quantum-scale particles behave like both particles and waves. Another new concept, the uncertainty principle, states that some of their properties taken together, such as their simultaneous position and momentum, in more recent times, wave–particle duality has been shown to apply not only to photons but to increasingly massive particles as well. Interactions of particles in the framework of field theory are understood as creation and annihilation of quanta of corresponding fundamental interactions. This blends particle physics with field theory, any subatomic particle, like any particle in the 3-dimensional space that obeys laws of quantum mechanics, can be either a boson or a fermion. Various extensions of the Standard Model predict the existence of a graviton particle. Composite subatomic particles are bound states of two or more elementary particles, for example, a proton is made of two up quarks and one down quark, while the atomic nucleus of helium-4 is composed of two protons and two neutrons. The neutron is made of two quarks and one up quark. Composite particles include all hadrons, these include baryons and mesons, in special relativity, the energy of a particle at rest equals its mass times the speed of light squared, E = mc2. That is, mass can be expressed in terms of energy, if a particle has a frame of reference where it lies at rest, then it has a positive rest mass and is referred to as massive. Baryons tend to have greater mass than mesons, which in turn tend to be heavier than leptons and it is also certain that any particle with an electric charge is massive. These include the photon and gluon, although the latter cannot be isolated, through the work of Albert Einstein, Satyendra Nath Bose, Louis de Broglie, and many others, current scientific theory holds that all particles also have a wave nature. This has been verified not only for elementary particles but also for compound particles like atoms, interactions between particles have been scrutinized for many centuries, and a few simple laws underpin how particles behave in collisions and interactions. These are the basics of Newtonian mechanics, a series of statements and equations in Philosophiae Naturalis Principia Mathematica. The negatively charged electron has an equal to 1⁄1837 or 1836 of that of a hydrogen atom
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Branch (mathematics)
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In mathematics, a principal branch is a function which selects one branch of a multi-valued function. Most often, this applies to functions defined on the complex plane, Principal branches are used in the definition of many inverse trigonometric functions, such as the selection either to define that arcsin, → or that arccos, →. A more familiar principal branch function, limited to numbers, is that of a positive real number raised to the power of 1/2. For example, take the relation y = x1/2, where x is any real number. This relation can be satisfied by any value of y equal to a root of x. By convention, √x is used to denote the positive root of x. In this instance, the square root function is taken as the principal branch of the multi-valued relation x1/2. One way to view a principal branch is to specifically at the exponential function. The exponential function is single-valued, where ez is defined as, however, the periodic nature of the trigonometric functions involved makes it clear that the logarithm is not so uniquely determined. Any number log z defined by such criteria has the property that elog z = z, in this manner log function is a multi-valued function. A branch cut, usually along the real axis, can limit the imaginary part so it lies between −π and π. These are the principal values. This is the branch of the log function. Often it is defined using a letter, Log z. Branch point Branch cut Complex logarithm Riemann surface Weisstein, Eric W. Principal Branch, branches of Complex Functions Module by John H. Mathews
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Asymptote
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In analytic geometry, an asymptote of a curve is a line such that the distance between the curve and the line approaches zero as they tend to infinity. Some sources include the requirement that the curve may not cross the line infinitely often, in some contexts, such as algebraic geometry, an asymptote is defined as a line which is tangent to a curve at infinity. The word asymptote is derived from the Greek ἀσύμπτωτος which means not falling together, + σύν together + πτωτ-ός fallen. The term was introduced by Apollonius of Perga in his work on conic sections, there are potentially three kinds of asymptotes, horizontal, vertical and oblique asymptotes. For curves given by the graph of a function y = ƒ, vertical asymptotes are vertical lines near which the function grows without bound. Asymptotes convey information about the behavior of curves in the large, the study of asymptotes of functions, construed in a broad sense, forms a part of the subject of asymptotic analysis. The idea that a curve may come close to a line without actually becoming the same may seem to counter everyday experience. The representations of a line and a curve as marks on a piece of paper or as pixels on a screen have a positive width. So if they were to be extended far enough they would seem to merge, but these are physical representations of the corresponding mathematical entities, the line and the curve are idealized concepts whose width is 0. Therefore, the understanding of the idea of an asymptote requires an effort of reason rather than experience, consider the graph of the function f =1 x shown to the right. The coordinates of the points on the curve are of the form where x is an other than 0. But no matter how large x becomes, its reciprocal 1 x is never 0, so the curve extends farther and farther upward as it comes closer and closer to the y-axis. Thus, both the x and y-axes are asymptotes of the curve and these ideas are part of the basis of concept of a limit in mathematics, and this connection is explained more fully below. The asymptotes most commonly encountered in the study of calculus are of curves of the form y = ƒ and these can be computed using limits and classified into horizontal, vertical and oblique asymptotes depending on its orientation. Horizontal asymptotes are lines that the graph of the function approaches as x tends to +∞ or −∞. As the name indicate they are parallel to the x-axis, vertical asymptotes are vertical lines near which the function grows without bound. Oblique asymptotes are diagonal lines so that the difference between the curve and the line approaches 0 as x tends to +∞ or −∞, more general type of asymptotes can be defined in this case. Only open curves that have some infinite branch, can have an asymptote, no closed curve can have an asymptote
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Symmetry
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Symmetry in everyday language refers to a sense of harmonious and beautiful proportion and balance. In mathematics, symmetry has a precise definition, that an object is invariant to any of various transformations. Although these two meanings of symmetry can sometimes be told apart, they are related, so they are discussed together. The opposite of symmetry is asymmetry, a geometric shape or object is symmetric if it can be divided into two or more identical pieces that are arranged in an organized fashion. This means that an object is symmetric if there is a transformation that moves individual pieces of the object, an object has rotational symmetry if the object can be rotated about a fixed point without changing the overall shape. An object has symmetry if it can be translated without changing its overall shape. An object has symmetry if it can be simultaneously translated and rotated in three-dimensional space along a line known as a screw axis. An object has symmetry if it does not change shape when it is expanded or contracted. Fractals also exhibit a form of symmetry, where small portions of the fractal are similar in shape to large portions. Other symmetries include glide reflection symmetry and rotoreflection symmetry, a dyadic relation R is symmetric if and only if, whenever its true that Rab, its true that Rba. Thus, is the age as is symmetrical, for if Paul is the same age as Mary. Symmetric binary logical connectives are and, or, biconditional, nand, xor, the set of operations that preserve a given property of the object form a group. In general, every kind of structure in mathematics will have its own kind of symmetry, examples include even and odd functions in calculus, the symmetric group in abstract algebra, symmetric matrices in linear algebra, and the Galois group in Galois theory. In statistics, it appears as symmetric probability distributions, and as skewness, symmetry in physics has been generalized to mean invariance—that is, lack of change—under any kind of transformation, for example arbitrary coordinate transformations. This concept has one of the most powerful tools of theoretical physics. See Noethers theorem, and also, Wigners classification, which says that the symmetries of the laws of physics determine the properties of the found in nature. Important symmetries in physics include continuous symmetries and discrete symmetries of spacetime, internal symmetries of particles, in biology, the notion of symmetry is mostly used explicitly to describe body shapes. Bilateral animals, including humans, are more or less symmetric with respect to the plane which divides the body into left
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Coordinate axes
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Each reference line is called a coordinate axis or just axis of the system, and the point where they meet is its origin, usually at ordered pair. The coordinates can also be defined as the positions of the projections of the point onto the two axis, expressed as signed distances from the origin. One can use the principle to specify the position of any point in three-dimensional space by three Cartesian coordinates, its signed distances to three mutually perpendicular planes. In general, n Cartesian coordinates specify the point in an n-dimensional Euclidean space for any dimension n and these coordinates are equal, up to sign, to distances from the point to n mutually perpendicular hyperplanes. The invention of Cartesian coordinates in the 17th century by René Descartes revolutionized mathematics by providing the first systematic link between Euclidean geometry and algebra. Using the Cartesian coordinate system, geometric shapes can be described by Cartesian equations, algebraic equations involving the coordinates of the points lying on the shape. For example, a circle of radius 2, centered at the origin of the plane, a familiar example is the concept of the graph of a function. Cartesian coordinates are also tools for most applied disciplines that deal with geometry, including astronomy, physics, engineering. They are the most common system used in computer graphics, computer-aided geometric design. Nicole Oresme, a French cleric and friend of the Dauphin of the 14th Century, used similar to Cartesian coordinates well before the time of Descartes. The adjective Cartesian refers to the French mathematician and philosopher René Descartes who published this idea in 1637 and it was independently discovered by Pierre de Fermat, who also worked in three dimensions, although Fermat did not publish the discovery. Both authors used a single axis in their treatments and have a length measured in reference to this axis. The concept of using a pair of axes was introduced later, after Descartes La Géométrie was translated into Latin in 1649 by Frans van Schooten and these commentators introduced several concepts while trying to clarify the ideas contained in Descartes work. Many other coordinate systems have developed since Descartes, such as the polar coordinates for the plane. The development of the Cartesian coordinate system would play a role in the development of the Calculus by Isaac Newton. The two-coordinate description of the plane was later generalized into the concept of vector spaces. Choosing a Cartesian coordinate system for a one-dimensional space – that is, for a straight line—involves choosing a point O of the line, a unit of length, and an orientation for the line. An orientation chooses which of the two half-lines determined by O is the positive, and which is negative, we say that the line is oriented from the negative half towards the positive half
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Eccentricity (mathematics)
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In mathematics, the eccentricity, denoted e or ε, is a parameter associated with every conic section. It can be thought of as a measure of how much the conic section deviates from being circular, in particular, The eccentricity of a circle is zero. The eccentricity of an ellipse which is not a circle is greater than zero, the eccentricity of a parabola is 1. The eccentricity of a hyperbola is greater than 1, the eccentricity of a line is infinite. Furthermore, two sections are similar if and only if they have the same eccentricity. Any conic section can be defined as the locus of points whose distances to a point and that ratio is called eccentricity, commonly denoted as e. The eccentricity can also be defined in terms of the intersection of a plane, for β =0 the plane section is a circle, for β = α a parabola. The linear eccentricity of an ellipse or hyperbola, denoted c, is the distance between its center and either of its two foci, the eccentricity can be defined as the ratio of the linear eccentricity to the semimajor axis a, that is, e = c a. The eccentricity is sometimes called first eccentricity to distinguish it from the second eccentricity, the eccentricity is also sometimes called numerical eccentricity. In the case of ellipses and hyperbolas the linear eccentricity is sometimes called half-focal separation, three notational conventions are in common use, e for the eccentricity and c for the linear eccentricity. ε for the eccentricity and e for the linear eccentricity, E or ϵ for the eccentricity and f for the linear eccentricity. This article uses the first notation, where, for the ellipse and the hyperbola, a is the length of the semi-major axis and b is the length of the semi-minor axis. The eccentricity of an ellipse is strictly less than 1, for any ellipse, let a be the length of its semi-major axis and b be the length of its semi-minor axis. The eccentricity is also the ratio of the axis a to the distance d from the center to the directrix. The eccentricity can be expressed in terms of the g, e = g. Define the maximum and minimum radii r max and r min as the maximum and minimum distances from either focus to the ellipse, the eccentricity of a hyperbola can be any real number greater than 1, with no upper bound. The eccentricity of a hyperbola is 2. The eccentricity of a quadric is the eccentricity of a designated section of it
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Focus (geometry)
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In geometry, focuses or foci, singular focus, are special points with reference to which any of a variety of curves is constructed. For example, one or two foci can be used in defining conic sections, the four types of which are the circle, ellipse, parabola, and hyperbola. In addition, two foci are used to define the Cassini oval and the Cartesian oval, and more than two foci are used in defining an n-ellipse. An ellipse can be defined as the 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 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 circle of Apollonius, 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 value of the difference between the distances to two given foci is a constant. It is also possible to describe all conic sections in terms of a focus and a single directrix. A conic is defined as the locus of points for each of which the distance to the focus divided by the distance to the directrix is a positive constant. If e is zero and one the conic is an ellipse, if e=1 the conic is a parabola. If the distance to the focus is fixed and the directrix is a line at infinity, so the eccentricity is zero and it is also possible to describe all the conic sections as loci of points that are equidistant from a single focus and a single, circular directrix. The ellipse thus generated has its focus at the center of the directrix circle. For the parabola, the center of the moves to the point at infinity. The directrix circle becomes a curve with zero curvature, indistinguishable from a straight line. To generate a hyperbola, the radius of the circle is chosen to be less than the distance between the center of this circle and the focus, thus, the focus is outside the directrix circle. The two branches of a hyperbola are thus the two halves of a curve closed over infinity, in projective geometry, all conics are equivalent in the sense that every theorem that can be stated for one can be stated for the others. Plutos ellipse is entirely inside Charons ellipse, as shown in animation of the system. The barycenter is about three-quarters of the distance from Earths center to its surface, moreover, the Pluto-Charon system moves in an ellipse around its barycenter with the Sun, as does the Earth-Moon system
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Paraboloid
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In geometry, a paraboloid is a quadric surface that has one axis of symmetry and no center of symmetry. The term paraboloid is derived from parabola, which refers to a section that has the same property of symmetry. In a suitable coordinate system with three axes x, y, and z, it can be represented by the equation z = x 2 a 2 + y 2 b 2. Where a and b are constants that dictate the level of curvature in the xz, in this position, the elliptic paraboloid opens upward. A hyperbolic paraboloid is a ruled surface shaped like a saddle. In a suitable system, a hyperbolic paraboloid can be represented by the equation z = y 2 b 2 − x 2 a 2. In this position, the hyperbolic paraboloid opens down along the x-axis, obviously both the paraboloids contain a lot of parabolas. But there are differences, too, an elliptic paraboloid contains ellipses. With a = b an elliptic paraboloid is a paraboloid of revolution and this shape is also called a circular paraboloid. This also works the way around, a parallel beam of light incident on the paraboloid parallel to its axis is concentrated at the focal point. This applies also for other waves, hence parabolic antennas, for a geometrical proof, click here. The hyperbolic paraboloid is a ruled surface, it contains two families of mutually skew lines. The lines in each family are parallel to a common plane, hence the hyperbolic paraboloid is a conoid. This property makes easy to realize a hyperbolic paraboloid with concrete, the widely sold fried snack food Pringles potato crisps resemble a truncated hyperbolic paraboloid. The distinctive shape of these allows them to be stacked in sturdy tubular containers. Examples in architecture St. a point, if the plane is a tangent plane, remark, an elliptic paraboloid is projectively equivalent to a sphere. Remarks, A hyperbolic paraboloid is a surface, but not developable. The Gauss curvature at any point is negative, hence it is a saddle surface
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Hyperboloid
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In geometry, a hyperboloid of revolution, sometimes called circular hyperboloid, is a surface that may be generated by rotating a hyperbola around one of its principal axes. An hyperboloid is a surface that may be obtained from a paraboloid of revolution by deforming it by means of directional scalings, or more generally, of an affine transformation. A hyperboloid is a surface, that is a surface that may be defined as the zero set of a polynomial of degree two in three variables. Among quadric surfaces, a hyperboloid is characterized by not being a cone or a cylinder, having a center of symmetry, a hyperboloid has also three pairwise perpendicular axes of symmetry, and three pairwise perpendicular planes of symmetry. Both of these surfaces are asymptotic to the cone of equation x 2 a 2 + y 2 b 2 − z 2 c 2 =0, one has an hyperboloid of revolution if and only if a 2 = b 2. It is a ruled surface. In case of a = b the hyperboloid is a surface of revolution, the more common generation of a hyperboloid of revolution is rotating a hyperbola around its semi-minor axis. Remark, A hyperboloid of two sheets is projectively equivalent to a hyperbolic paraboloid, for simplicity the plane sections of the unit hyperboloid with equation H1, x 2 + y 2 − z 2 =1 are considered. Because a hyperboloid in general position is an image of the unit hyperboloid. The hyperboloid of two sheets does not contain lines, the discussion of plane sections can be performed for the unit hyperboloid of two sheets with equation H2, x 2 + y 2 − z 2 = −1. Remark, A hyperboloid of two sheets is projectively equivalent to a sphere, whereas the Gaussian curvature of a hyperboloid of one sheet is negative, that of a two-sheet hyperboloid is positive. In spite of its curvature, the hyperboloid of two sheets with another suitably chosen metric can also be used as a model for hyperbolic geometry. More generally, an arbitrarily oriented hyperboloid, centered at v, is defined by the equation T A =1, where A is a matrix and x, v are vectors. The eigenvectors of A define the directions of the hyperboloid. The one-sheet hyperboloid has two positive eigenvalues and one negative eigenvalue, the two-sheet hyperboloid has one positive eigenvalue and two negative eigenvalues. Imaginary hyperboloids are frequently found in mathematics of higher dimensions, for example, in a pseudo-Euclidean space one has the use of a quadratic form, q = −, k < n. When c is any constant, then the part of the space given by is called a hyperboloid, the degenerate case corresponds to c =0. As an example, consider the following passage, however, the term quasi-sphere is also used in this context since the sphere and hyperboloid have some commonality
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Hyperbolic geometry
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In mathematics, hyperbolic geometry is a non-Euclidean geometry. Hyperbolic plane geometry is also the geometry of saddle surface or pseudospherical surfaces, surfaces with a constant negative Gaussian curvature, a modern use of hyperbolic geometry is in the theory of special relativity, particularly Minkowski spacetime and gyrovector space. In Russia it is commonly called Lobachevskian geometry, named one of its discoverers. This page is mainly about the 2-dimensional hyperbolic geometry and the differences and similarities between Euclidean and hyperbolic geometry, Hyperbolic geometry can be extended to three and more dimensions, see hyperbolic space for more on the three and higher dimensional cases. Hyperbolic geometry is closely related to Euclidean geometry than it seems. When the parallel postulate is removed from Euclidean geometry the resulting geometry is absolute geometry, there are two kinds of absolute geometry, Euclidean and hyperbolic. All theorems of geometry, including the first 28 propositions of book one of Euclids Elements, are valid in Euclidean. Propositions 27 and 28 of Book One of Euclids Elements prove the existence of parallel/non-intersecting lines and this difference also has many consequences, concepts that are equivalent in Euclidean geometry are not equivalent in hyperbolic geometry, new concepts need to be introduced. Further, because of the angle of parallelism hyperbolic geometry has an absolute scale, single lines in hyperbolic geometry have exactly the same properties as single straight lines in Euclidean geometry. For example, two points define a line, and lines can be infinitely extended. Two intersecting lines have the properties as two intersecting lines in Euclidean geometry. For example, two lines can intersect in no more than one point, intersecting lines have equal opposite angles, when we add a third line then there are properties of intersecting lines that differ from intersecting lines in Euclidean geometry. For example, given 2 intersecting lines there are many lines that do not intersect either of the given lines. While in some models lines look different they do have these properties, non-intersecting lines in hyperbolic geometry also have properties that differ from non-intersecting lines in Euclidean geometry, For any line R and any point P which does not lie on R. In the plane containing line R and point P there are at least two lines through P that do not intersect R. This implies that there are through P an infinite number of lines that do not intersect R. All other non-intersecting lines have a point of distance and diverge from both sides of that point, and are called ultraparallel, diverging parallel or sometimes non-intersecting. Some geometers simply use parallel lines instead of limiting parallel lines and these limiting parallels make an angle θ with PB, this angle depends only on the Gaussian curvature of the plane and the distance PB and is called the angle of parallelism
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Nikolai Lobachevsky
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Nikolai Ivanovich Lobachevsky was a Russian mathematician and geometer, known primarily for his work on hyperbolic geometry, otherwise known as Lobachevskian geometry. William Kingdon Clifford called Lobachevsky the Copernicus of Geometry due to the character of his work. He was one of three children and his father, a clerk in a land surveying office, died when he was seven, and his mother moved to Kazan. Lobachevsky attended Kazan Gymnasium from 1802, graduating in 1807 and then received a scholarship to Kazan University, at Kazan University, Lobachevsky was influenced by professor Johann Christian Martin Bartels, a former teacher and friend of German mathematician Carl Friedrich Gauss. Lobachevsky received a degree in physics and mathematics in 1811. He served in administrative positions and became the rector of Kazan University in 1827. In 1832, he married Varvara Alexeyevna Moiseyeva and they had a large number of children. He was dismissed from the university in 1846, ostensibly due to his health, by the early 1850s, he was nearly blind. He died in poverty in 1856, Lobachevskys main achievement is the development of a non-Euclidean geometry, also referred to as Lobachevskian geometry. Before him, mathematicians were trying to deduce Euclids fifth postulate from other axioms, Euclids fifth is a rule in Euclidean geometry which states that for any given line and point not on the line, there is one parallel line through the point not intersecting the line. Lobachevsky would instead develop a geometry in which the fifth postulate was not true and this idea was first reported on February 23,1826 to the session of the department of physics and mathematics, and this research was printed in the UMA in 1829–1830. The non-Euclidean geometry that Lobachevsky developed is referred to as hyperbolic geometry and he developed the angle of parallelism which depends on the distance the point is off the given line. In hyperbolic geometry the sum of angles in a triangle must be less than 180 degrees. Non-Euclidean geometry stimulated the development of geometry which has many applications. Hyperbolic geometry is referred to as Lobachevskian geometry or Bolyai–Lobachevskian geometry. Some mathematicians and historians have claimed that Lobachevsky in his studies in non-Euclidean geometry was influenced by Gauss. Gauss himself appreciated Lobachevskys published works very highly, but they never had personal correspondence between them prior to the publication, Lobachevskys magnum opus Geometriya was completed in 1823, but was not published in its exact original form until 1909, long after he had died. Lobachevsky was also the author of New Foundations of Geometry and he also wrote Geometrical Investigations on the Theory of Parallels and Pangeometry
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Non-Euclidean geometry
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In mathematics, non-Euclidean geometry consists of two geometries based on axioms closely related to those specifying Euclidean geometry. In the latter case one obtains hyperbolic geometry and elliptic geometry, when the metric requirement is relaxed, then there are affine planes associated with the planar algebras which give rise to kinematic geometries that have also been called non-Euclidean geometry. The essential difference between the geometries is the nature of parallel lines. In hyperbolic geometry, by contrast, there are many lines through A not intersecting ℓ, while in elliptic geometry. In elliptic geometry the lines curve toward each other and intersect, the debate that eventually led to the discovery of the non-Euclidean geometries began almost as soon as Euclids work Elements was written. In the Elements, Euclid began with a number of assumptions. Other mathematicians have devised simpler forms of this property, regardless of the form of the postulate, however, it consistently appears to be more complicated than Euclids other postulates,1. To draw a line from any point to any point. To produce a straight line continuously in a straight line. To describe a circle with any centre and distance and that all right angles are equal to one another. For at least a thousand years, geometers were troubled by the complexity of the fifth postulate. Many attempted to find a proof by contradiction, including Ibn al-Haytham, Omar Khayyám, Nasīr al-Dīn al-Tūsī and these theorems along with their alternative postulates, such as Playfairs axiom, played an important role in the later development of non-Euclidean geometry. These early attempts did, however, provide some early properties of the hyperbolic and elliptic geometries. Another example is al-Tusis son, Sadr al-Din, who wrote a book on the subject in 1298, based on al-Tusis later thoughts and he essentially revised both the Euclidean system of axioms and postulates and the proofs of many propositions from the Elements. His work was published in Rome in 1594 and was studied by European geometers and he finally reached a point where he believed that his results demonstrated the impossibility of hyperbolic geometry. His claim seems to have based on Euclidean presuppositions, because no logical contradiction was present. In this attempt to prove Euclidean geometry he instead unintentionally discovered a new viable geometry, in 1766 Johann Lambert wrote, but did not publish, Theorie der Parallellinien in which he attempted, as Saccheri did, to prove the fifth postulate. He worked with a figure that today we call a Lambert quadrilateral and he quickly eliminated the possibility that the fourth angle is obtuse, as had Saccheri and Khayyam, and then proceeded to prove many theorems under the assumption of an acute angle
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Hyperbolic function
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In mathematics, hyperbolic functions are analogs of the ordinary trigonometric, or circular functions. The inverse hyperbolic functions are the hyperbolic sine arsinh and so on. Just as the form a circle with a unit radius. The hyperbolic functions take a real argument called a hyperbolic angle, the size of a hyperbolic angle is twice the area of its hyperbolic sector. The hyperbolic functions may be defined in terms of the legs of a triangle covering this sector. Laplaces equations are important in areas of physics, including electromagnetic theory, heat transfer, fluid dynamics. In complex analysis, the hyperbolic functions arise as the parts of sine and cosine. When considered defined by a variable, the hyperbolic functions are rational functions of exponentials. Hyperbolic functions were introduced in the 1760s independently by Vincenzo Riccati, Riccati used Sc. and Cc. to refer to circular functions and Sh. and Ch. to refer to hyperbolic functions. Lambert adopted the names but altered the abbreviations to what they are today, the abbreviations sh and ch are still used in some other languages, like French and Russian. The hyperbolic functions are, Hyperbolic sine, sinh x = e x − e − x 2 = e 2 x −12 e x =1 − e −2 x 2 e − x. Hyperbolic cosine, cosh x = e x + e − x 2 = e 2 x +12 e x =1 + e −2 x 2 e − x, the complex forms in the definitions above derive from Eulers formula. One also has sech 2 x =1 − tanh 2 x csch 2 x = coth 2 x −1 for the other functions, sinh = sinh 2 = sgn cosh −12 where sgn is the sign function. All functions with this property are linear combinations of sinh and cosh, in particular the exponential functions e x and e − x, and it is possible to express the above functions as Taylor series, sinh x = x + x 33. + ⋯ = ∑ n =0 ∞ x 2 n +1, the function sinh x has a Taylor series expression with only odd exponents for x. Thus it is an odd function, that is, −sinh x = sinh, the function cosh x has a Taylor series expression with only even exponents for x. Thus it is a function, that is, symmetric with respect to the y-axis. The sum of the sinh and cosh series is the series expression of the exponential function
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Gyrovector space
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A gyrovector space is a mathematical concept proposed by Abraham A. Ungar for studying hyperbolic geometry in analogy to the way vector spaces are used in Euclidean geometry. Ungar introduced the concept of gyrovectors that have addition based on gyrogroups instead of vectors which have addition based on groups, Ungar developed his concept as a tool for the formulation of special relativity as an alternative to the use of Lorentz transformations to represent compositions of velocities. This is achieved by introducing gyro operators, two 3d velocity vectors are used to construct an operator, which acts on another 3d velocity, Gyrogroups are a type of Bol loop. Gyrocommutative gyrogroups are equivalent to K-loops although defined differently, the terms Bruck loop and dyadic symset are also in use. A groupoid is a gyrogroup if its binary operation satisfies the following axioms, for each a ∈ G there is an element ⊖ a in G called a left inverse of a with ⊖ a ⊕ a =0. That is gyr is a member of Aut and the automorphism gyr of G is called the gyroautomorphism of G generated by a, b in G, the operation gyr, G × G → Aut is called the gyrator of G. The gyroautomorphism gyr has the left loop property gyr = gyr The first pair of axioms are like the group axioms, the last pair present the gyrator axioms and the middle axiom links the two pairs. Since a gyrogroup has inverses and an identity it qualifies as a quasigroup, Gyrogroups are a generalization of groups. Every group is an example of a gyrogroup with gyr defined as the identity map, an example of a finite gyrogroup is given in. Some identities which hold in any gyrogroup, g y r w = ⊖ ⊕ u ⊕ = ⊕ g y r w ⊕ w = u ⊕ More identities given on page 50 of. A gyrogroup is gyrocommutative if its binary operation obeys the gyrocommutative law, coaddition is commutative if the gyrogroup addition is gyrocommutative. Einstein velocity addition is commutative and associative only when u and v are parallel, if the 3 ×3 matrix form of the rotation applied to 3-coordinates is given by gyr, then the 4 ×4 matrix rotation applied to 4-coordinates is given by, G y r =. The composition of two Lorentz transformations L and L which include rotations U and V is given by, L L = L In the above, the matrix entries depend on the components of the 3-velocity v, and thats what the notation B means. Let s be any constant, let be any real inner product space. Let s be any constant, let be any real inner product space. This formula provides a model that uses a whole compared to other models of hyperbolic geometry which use discs or half-planes. A gyrovector space isomorphism preserves gyrogroup addition and scalar multiplication and the inner product, the three gyrovector spaces Möbius, Einstein and Proper Velocity are isomorphic. Gyrotrigonometry is the use of gyroconcepts to study hyperbolic triangles, gyrotrigonometry takes the approach of using the ordinary trigonometric functions but in conjunction with gyrotriangle identities
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Theory of relativity
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The theory of relativity usually encompasses two interrelated theories by Albert Einstein, special relativity and general relativity. Special relativity applies to particles and their interactions, describing all their physical phenomena except gravity. General relativity explains the law of gravitation and its relation to other forces of nature and it applies to the cosmological and astrophysical realm, including astronomy. The theory transformed theoretical physics and astronomy during the 20th century and it introduced concepts including spacetime as a unified entity of space and time, relativity of simultaneity, kinematic and gravitational time dilation, and length contraction. In the field of physics, relativity improved the science of elementary particles and their fundamental interactions, with relativity, cosmology and astrophysics predicted extraordinary astronomical phenomena such as neutron stars, black holes, and gravitational waves. Max Planck, Hermann Minkowski and others did subsequent work, Einstein developed general relativity between 1907 and 1915, with contributions by many others after 1915. The final form of general relativity was published in 1916, the term theory of relativity was based on the expression relative theory used in 1906 by Planck, who emphasized how the theory uses the principle of relativity. In the discussion section of the paper, Alfred Bucherer used for the first time the expression theory of relativity. By the 1920s, the community understood and accepted special relativity. It rapidly became a significant and necessary tool for theorists and experimentalists in the new fields of physics, nuclear physics. By comparison, general relativity did not appear to be as useful and it seemed to offer little potential for experimental test, as most of its assertions were on an astronomical scale. Its mathematics of general relativity seemed difficult and fully understandable only by a number of people. Around 1960, general relativity became central to physics and astronomy, new mathematical techniques to apply to general relativity streamlined calculations and made its concepts more easily visualized. Special relativity is a theory of the structure of spacetime and it was introduced in Einsteins 1905 paper On the Electrodynamics of Moving Bodies. Special relativity is based on two postulates which are contradictory in classical mechanics, The laws of physics are the same for all observers in motion relative to one another. The speed of light in a vacuum is the same for all observers, the resultant theory copes with experiment better than classical mechanics. For instance, postulate 2 explains the results of the Michelson–Morley experiment, moreover, the theory has many surprising and counterintuitive consequences. Some of these are, Relativity of simultaneity, Two events, simultaneous for one observer, time dilation, Moving clocks are measured to tick more slowly than an observers stationary clock
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Quantum mechanics
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Quantum mechanics, including quantum field theory, is a branch of physics which is the fundamental theory of nature at small scales and low energies of atoms and subatomic particles. Classical physics, the physics existing before quantum mechanics, derives from quantum mechanics as an approximation valid only at large scales, early quantum theory was profoundly reconceived in the mid-1920s. The reconceived theory is formulated in various specially developed mathematical formalisms, in one of them, a mathematical function, the wave function, provides information about the probability amplitude of position, momentum, and other physical properties of a particle. In 1803, Thomas Young, an English polymath, performed the famous experiment that he later described in a paper titled On the nature of light. This experiment played a role in the general acceptance of the wave theory of light. In 1838, Michael Faraday discovered cathode rays, Plancks hypothesis that energy is radiated and absorbed in discrete quanta precisely matched the observed patterns of black-body radiation. In 1896, Wilhelm Wien empirically determined a distribution law of black-body radiation, ludwig Boltzmann independently arrived at this result by considerations of Maxwells equations. However, it was only at high frequencies and underestimated the radiance at low frequencies. Later, Planck corrected this model using Boltzmanns statistical interpretation of thermodynamics and proposed what is now called Plancks law, following Max Plancks solution in 1900 to the black-body radiation problem, Albert Einstein offered a quantum-based theory to explain the photoelectric effect. Among the first to study quantum phenomena in nature were Arthur Compton, C. V. Raman, robert Andrews Millikan studied the photoelectric effect experimentally, and Albert Einstein developed a theory for it. In 1913, Peter Debye extended Niels Bohrs theory of structure, introducing elliptical orbits. This phase is known as old quantum theory, according to Planck, each energy element is proportional to its frequency, E = h ν, where h is Plancks constant. Planck cautiously insisted that this was simply an aspect of the processes of absorption and emission of radiation and had nothing to do with the reality of the radiation itself. In fact, he considered his quantum hypothesis a mathematical trick to get the right rather than a sizable discovery. He won the 1921 Nobel Prize in Physics for this work, lower energy/frequency means increased time and vice versa, photons of differing frequencies all deliver the same amount of action, but do so in varying time intervals. High frequency waves are damaging to human tissue because they deliver their action packets concentrated in time, the Copenhagen interpretation of Niels Bohr became widely accepted. In the mid-1920s, developments in mechanics led to its becoming the standard formulation for atomic physics. In the summer of 1925, Bohr and Heisenberg published results that closed the old quantum theory, out of deference to their particle-like behavior in certain processes and measurements, light quanta came to be called photons
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Euclidean geometry
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Euclidean geometry is a mathematical system attributed to the Alexandrian Greek mathematician Euclid, which he described in his textbook on geometry, the Elements. Euclids method consists in assuming a set of intuitively appealing axioms. Although many of Euclids results had been stated by earlier mathematicians, Euclid was the first to show how these propositions could fit into a comprehensive deductive and logical system. The Elements begins with plane geometry, still taught in school as the first axiomatic system. It goes on to the geometry of three dimensions. Much of the Elements states results of what are now called algebra and number theory, for more than two thousand years, the adjective Euclidean was unnecessary because no other sort of geometry had been conceived. Euclids axioms seemed so obvious that any theorem proved from them was deemed true in an absolute, often metaphysical. Today, however, many other self-consistent non-Euclidean geometries are known, Euclidean geometry is an example of synthetic geometry, in that it proceeds logically from axioms to propositions without the use of coordinates. This is in contrast to analytic geometry, which uses coordinates, the Elements is mainly a systematization of earlier knowledge of geometry. Its improvement over earlier treatments was recognized, with the result that there was little interest in preserving the earlier ones. There are 13 total books in the Elements, Books I–IV, Books V and VII–X deal with number theory, with numbers treated geometrically via their representation as line segments with various lengths. Notions such as numbers and rational and irrational numbers are introduced. The infinitude of prime numbers is proved, a typical result is the 1,3 ratio between the volume of a cone and a cylinder with the same height and base. Euclidean geometry is a system, in which all theorems are derived from a small number of axioms. To produce a straight line continuously in a straight line. To describe a circle with any centre and distance and that all right angles are equal to one another. Although Euclids statement of the only explicitly asserts the existence of the constructions. The Elements also include the five common notions, Things that are equal to the same thing are also equal to one another
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Greek language
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Greek is an independent branch of the Indo-European family of languages, native to Greece and other parts of the Eastern Mediterranean. It has the longest documented history of any living language, spanning 34 centuries of written records and its writing system has been the Greek alphabet for the major part of its history, other systems, such as Linear B and the Cypriot syllabary, were used previously. The alphabet arose from the Phoenician script and was in turn the basis of the Latin, Cyrillic, Armenian, Coptic, Gothic and many other writing systems. Together with the Latin texts and traditions of the Roman world, during antiquity, Greek was a widely spoken lingua franca in the Mediterranean world and many places beyond. It would eventually become the official parlance of the Byzantine Empire, the language is spoken by at least 13.2 million people today in Greece, Cyprus, Italy, Albania, Turkey, and the Greek diaspora. Greek roots are used to coin new words for other languages, Greek. Greek has been spoken in the Balkan peninsula since around the 3rd millennium BC, the earliest written evidence is a Linear B clay tablet found in Messenia that dates to between 1450 and 1350 BC, making Greek the worlds oldest recorded living language. Among the Indo-European languages, its date of earliest written attestation is matched only by the now extinct Anatolian languages, the Greek language is conventionally divided into the following periods, Proto-Greek, the unrecorded but assumed last ancestor of all known varieties of Greek. The unity of Proto-Greek would have ended as Hellenic migrants entered the Greek peninsula sometime in the Neolithic era or the Bronze Age, Mycenaean Greek, the language of the Mycenaean civilisation. It is recorded in the Linear B script on tablets dating from the 15th century BC onwards, Ancient Greek, in its various dialects, the language of the Archaic and Classical periods of the ancient Greek civilisation. It was widely known throughout the Roman Empire, after the Roman conquest of Greece, an unofficial bilingualism of Greek and Latin was established in the city of Rome and Koine Greek became a first or second language in the Roman Empire. The origin of Christianity can also be traced through Koine Greek, Medieval Greek, also known as Byzantine Greek, the continuation of Koine Greek in Byzantine Greece, up to the demise of the Byzantine Empire in the 15th century. Much of the written Greek that was used as the language of the Byzantine Empire was an eclectic middle-ground variety based on the tradition of written Koine. Modern Greek, Stemming from Medieval Greek, Modern Greek usages can be traced in the Byzantine period and it is the language used by the modern Greeks, and, apart from Standard Modern Greek, there are several dialects of it. In the modern era, the Greek language entered a state of diglossia, the historical unity and continuing identity between the various stages of the Greek language is often emphasised. Greek speakers today still tend to regard literary works of ancient Greek as part of their own rather than a foreign language and it is also often stated that the historical changes have been relatively slight compared with some other languages. According to one estimation, Homeric Greek is probably closer to demotic than 12-century Middle English is to modern spoken English, Greek is spoken by about 13 million people, mainly in Greece, Albania and Cyprus, but also worldwide by the large Greek diaspora. Greek is the language of Greece, where it is spoken by almost the entire population
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Doubling the cube
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Doubling the cube, also known as the Delian problem, is an ancient geometric problem. Given the edge of a cube, the problem requires the construction of the edge of a cube whose volume is double that of the first, using only the tools of a compass. As with the problems of squaring the circle and trisecting the angle. The Egyptians, Indians, and particularly the Greeks were aware of the problem and made futile attempts at solving what they saw as an obstinate. However, the nonexistence of a solution was finally proven by Pierre Wantzel in 1837, in algebraic terms, doubling a unit cube requires the construction of a line segment of length x, where x3 =2, in other words, x = 3√2. This is because a cube of side length 1 has a volume of 13 =1, the impossibility of doubling the cube is therefore equivalent to the statement that 3√2 is not a constructible number. This implies that the degree of the extension generated by a constructible point must be a power of 2. The field extension generated by 3√2, however, is of degree 3 and we begin with the unit line segment defined by points and in the plane. We are required to construct a line segment defined by two separated by a distance of 3√2. Any newly defined point either arises as the result of the intersection of two circles, as the intersection of a circle and a line, or as the intersection of two lines. Restated in more abstract terminology, the new x- and y-coordinates have minimal polynomials of degree at most 2 over the subfield of ℝ generated by the previous coordinates, therefore, the degree of the field extension corresponding to each new coordinate is 2 or 1. By Gausss Lemma, p is irreducible over ℚ, and is thus a minimal polynomial over ℚ for 3√2. The field extension ℚ, ℚ is therefore of degree 3. But this is not a power of 2, so by the above, 3√2 is not the coordinate of a point, and thus a line segment of 3√2 cannot be constructed. The problem owes its name to a story concerning the citizens of Delos, the oracle responded that they must double the size of the altar to Apollo, which was a regular cube. This may be why the problem is referred to in the 350s BC by the author of the pseudo-Platonic Sisyphus as still unsolved, however another version of the story says that all three found solutions but they were too abstract to be of practical value. In modern notation, this means that given segments of lengths a and 2a, the duplication of the cube is equivalent to finding segments of lengths r and s so that a r = r s = s 2 a. In turn, this means that r = a ⋅23 But Pierre Wantzel proved in 1837 that the root of 2 is not constructible
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Apollonius of Perga
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Apollonius of Perga was a Greek geometer and astronomer known for his theories on the topic of conic sections. Beginning from the theories of Euclid and Archimedes on the topic and his definitions of the terms ellipse, parabola, and hyperbola are the ones in use today. Apollonius worked on other topics, including astronomy. Most of the work has not survived except in references in other authors. His hypothesis of eccentric orbits to explain the apparently aberrant motion of the planets, for such an important contributor to the field of mathematics, scant biographical information remains. The 6th century Palestinian commentator, Eutocius of Ascalon, on Apollonius’ major work, Conics, states, “Apollonius, the geometrician. Came from Perga in Pamphylia in the times of Ptolemy Euergetes, the ruins of the city yet stand. It was a center of Hellenistic culture, Euergetes, “benefactor, ” identifies Ptolemy III Euergetes, third Greek dynast of Egypt in the diadochi succession. Presumably, his “times” are his regnum, 246-222/221 BC, times are always recorded by ruler or officiating magistrate, so that if Apollonius was born earlier than 246, it would have been the “times” of Euergetes’ father. The identity of Herakleios is uncertain, the approximate times of Apollonius are thus certain, but no exact dates can be given. The figure Specific birth and death years stated by the scholars are only speculative. Eutocius appears to associate Perga with the Ptolemaic dynasty of Egypt, never under Egypt, Perga in 246 BC belonged to the Seleucid Empire, an independent diadochi state ruled by the Seleucid dynasty. Someone designated “of Perga” might well be expected to have lived and worked there, to the contrary, if Apollonius was later identified with Perga, it was not on the basis of his residence. The remaining autobiographical material implies that he lived, studied and wrote in Alexandria, philip was assassinated in 336 BC. Alexander went on to fulfill his plan by conquering the vast Iranian empire, the material is located in the surviving false “Prefaces” of the books of his Conics. These are letters delivered to friends of Apollonius asking them to review the book enclosed with the letter. The Preface to Book I, addressed to one Eudemus, reminds him that Conics was initially requested by a house guest at Alexandria, Naucrates had the first draft of all eight books in his hands by the end of the visit. Apollonius refers to them as being “without a thorough purgation” and he intended to verify and emend the books, releasing each one as it was completed
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Locus (mathematics)
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In geometry, a locus is a set of points, whose location satisfies or is determined by one or more specified conditions. Until the beginning of 20th century, a shape was not considered as an infinite set of points, rather. Thus a circle in the Euclidean plane was defined as the locus of a point that is at a distance of a fixed point. In contrast to the view, the old formulation avoids considering infinite collections. Once set theory became the universal basis over which the mathematics is built. Examples from plane geometry include, The set of points equidistant from two points is a perpendicular bisector to the segment connecting the two points. The set of points equidistant from two lines cross is the angle bisector. All conic sections are loci, Parabola, the set of points equidistant from a single point, Circle, the set of points for which the distance from a single point is constant. The set of points for each of which the ratio of the distances to two given foci is a constant is referred to as a Circle of Apollonius. Hyperbola, the set of points for each of which the value of the difference between the distances to two given foci is a constant. Ellipse, the set of points for each of which the sum of the distances to two given foci is a constant, the circle is the special case in which the two foci coincide with each other. Other examples of loci appear in areas of mathematics. For example, in dynamics, the Mandelbrot set is a subset of the complex plane that may be characterized as the connectedness locus of a family of polynomial maps. Proof that all the points on the given shape satisfy the conditions and we find the locus of the points P that have a given ratio of distances k = d1/d2 to two given points. In this example we choose k=3, A and B as the fixed points and it is the circle of Apollonius defined by these values of k, A, and B. A triangle ABC has a side with length c. We determine the locus of the third vertex C such that the medians from A and C are orthogonal and we choose an orthonormal coordinate system such that A, B. C is the third vertex
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Congruence (geometry)
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In geometry, two figures or objects are congruent if they have the same shape and size, or if one has the same shape and size as the mirror image of the other. This means that either object can be repositioned and reflected so as to coincide precisely with the other object, so two distinct plane figures on a piece of paper are congruent if we can cut them out and then match them up completely. Turning the paper over is permitted, in elementary geometry the word congruent is often used as follows. The word equal is often used in place of congruent for these objects, two line segments are congruent if they have the same length. Two angles are congruent if they have the same measure, two circles are congruent if they have the same diameter. The related concept of similarity applies if the objects differ in size, for two polygons to be congruent, they must have an equal number of sides. Two polygons with n sides are congruent if and only if they each have identical sequences side-angle-side-angle-. for n sides. Congruence of polygons can be established graphically as follows, First, match, second, draw a vector from one of the vertices of the one of the figures to the corresponding vertex of the other figure. Translate the first figure by this vector so that two vertices match. Third, rotate the translated figure about the matched vertex until one pair of corresponding sides matches, fourth, reflect the rotated figure about this matched side until the figures match. If at any time the step cannot be completed, the polygons are not congruent, two triangles are congruent if their corresponding sides are equal in length, in which case their corresponding angles are equal in measure. SSS, If three pairs of sides of two triangles are equal in length, then the triangles are congruent, ASA, If two pairs of angles of two triangles are equal in measurement, and the included sides are equal in length, then the triangles are congruent. The ASA Postulate was contributed by Thales of Miletus, in most systems of axioms, the three criteria—SAS, SSS and ASA—are established as theorems. In the School Mathematics Study Group system SAS is taken as one of 22 postulates, AAS, If two pairs of angles of two triangles are equal in measurement, and a pair of corresponding non-included sides are equal in length, then the triangles are congruent. For American usage, AAS is equivalent to an ASA condition, RHS, also known as HL, If two right-angled triangles have their hypotenuses equal in length, and a pair of shorter sides are equal in length, then the triangles are congruent. The SSA condition which specifies two sides and a non-included angle does not by itself prove congruence, in order to show congruence, additional information is required such as the measure of the corresponding angles and in some cases the lengths of the two pairs of corresponding sides. The opposite side is longer when the corresponding angles are acute. This is the case and two different triangles can be formed from the given information, but further information distinguishing them can lead to a proof of congruence
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Symmetry (geometry)
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A geometric object has symmetry if there is an operation or transformation that maps the figure/object onto itself, i. e. it is said that the object has an invariance under the transform. For instance, a circle rotated about its center will have the shape and size as the original circle—all points before. A circle is said to be symmetric under rotation or to have rotational symmetry, the types of symmetries that are possible for a geometric object depend on the set of geometric transforms available, and on what object properties should remain unchanged after a transform. These isometries consist of reflections, rotations, translations, and combinations of basic operations. Under an isometric transformation, an object is said to be symmetric if, after transformation. A geometric object is symmetric only under a subset or subgroup of all isometries. The kinds of isometry subgroups are described below, followed by other kinds of transform groups, two generating mirrors of a striation create infinitely many virtual copies following a horocycle. Reflectional symmetry, linear symmetry, mirror symmetry, mirror-image symmetry, in one dimension, there is a point of symmetry about which reflection takes place, in two dimensions there is an axis of symmetry, and in three dimensions there is a plane of symmetry. An object or figure which is indistinguishable from its image is called mirror symmetric. Another way to think about it is if the shape were to be folded in half over the axis. Thus a square has four axis of symmetry, because there are four different ways to fold it and have the edges all match, a circle has infinitely many axis of symmetry passing through its center, for the same reason. If the letter T is reflected along an axis, it appears the same. This is sometimes called vertical symmetry, one can better use an unambiguous formulation, e. g. T has a vertical symmetry axis or T has left-right symmetry. The triangles with reflection symmetry are isosceles, the quadrilaterals with this symmetry are the kites, for each line or plane of reflection, the symmetry group is isomorphic with Cs, one of the three types of order two, hence algebraically isomorphic to C2. The fundamental domain is a half-plane or half-space, reflection symmetry can be generalized to other isometries of m-dimensional space which are involutions, such as ↦ in a certain system of Cartesian coordinates. This reflects the space along an -dimensional affine subspace, If k = m, then such a transformation is known as a point reflection, or an inversion through a point. On the plane a point reflection is the same as a half-turn rotation, antipodal symmetry is an alternative name for a point reflection symmetry through the origin. Such a reflection preserves orientation if and only if k is an even number and this implies that for m =3 a point reflection changes the orientation of the space, like a mirror-image symmetry
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Similarity (geometry)
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Two geometrical objects are called similar if they both have the same shape, or one has the same shape as the mirror image of the other. More precisely, one can be obtained from the other by uniformly scaling and this means that either object can be rescaled, repositioned, and reflected, so as to coincide precisely with the other object. If two objects are similar, each is congruent to the result of a uniform scaling of the other. A modern and novel perspective of similarity is to consider geometrical objects similar if one appears congruent to the other zoomed in or out at some level. For example, all circles are similar to other, all squares are similar to each other. On the other hand, ellipses are not all similar to other, rectangles are not all similar to each other. If two angles of a triangle have measures equal to the measures of two angles of triangle, then the triangles are similar. Corresponding sides of similar polygons are in proportion, and corresponding angles of similar polygons have the same measure and it can be shown that two triangles having congruent angles are similar, that is, the corresponding sides can be proved to be proportional. This is known as the AAA similarity theorem, due to this theorem, several authors simplify the definition of similar triangles to only require that the corresponding three angles are congruent. There are several statements each of which is necessary and sufficient for two triangles to be similar,1, the triangles have two congruent angles, which in Euclidean geometry implies that all their angles are congruent. That is, If ∠BAC is equal in measure to ∠B′A′C′, and ∠ABC is equal in measure to ∠A′B′C′, then this implies that ∠ACB is equal in measure to ∠A′C′B′, all the corresponding sides have lengths in the same ratio, AB/A′B′ = BC/B′C′ = AC/A′C′. This is equivalent to saying that one triangle is an enlargement of the other, two sides have lengths in the same ratio, and the angles included between these sides have the same measure. For instance, AB/A′B′ = BC/B′C′ and ∠ABC is equal in measure to ∠A′B′C′ and this is known as the SAS Similarity Criterion. When two triangles △ABC and △A′B′C′ are similar, one writes △ABC ∼ △A′B′C′, there are several elementary results concerning similar triangles in Euclidean geometry, Any two equilateral triangles are similar. Two triangles, both similar to a triangle, are similar to each other. Corresponding altitudes of similar triangles have the ratio as the corresponding sides. Two right triangles are similar if the hypotenuse and one side have lengths in the same ratio. Given a triangle △ABC and a line segment DE one can, with ruler and compass, the statement that the point F satisfying this condition exists is Walliss Postulate and is logically equivalent to Euclids Parallel Postulate
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Translation (geometry)
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In Euclidean geometry, a translation is a geometric transformation that moves every point of a figure or a space by the same amount in a given direction. In Euclidean geometry a transformation is a correspondence between two sets of points or a mapping from one plane to another. )A translation can be described as a rigid motion. A translation can also be interpreted as the addition of a constant vector to every point, a translation operator is an operator T δ such that T δ f = f. If v is a vector, then the translation Tv will work as Tv. If T is a translation, then the image of a subset A under the function T is the translate of A by T, the translate of A by Tv is often written A + v. In a Euclidean space, any translation is an isometry, the set of all translations forms the translation group T, which is isomorphic to the space itself, and a normal subgroup of Euclidean group E. The quotient group of E by T is isomorphic to the orthogonal group O, E / T ≅ O, a translation is an affine transformation with no fixed points. Matrix multiplications always have the origin as a fixed point, similarly, the product of translation matrices is given by adding the vectors, T u T v = T u + v. Because addition of vectors is commutative, multiplication of matrices is therefore also commutative. In physics, translation is movement that changes the position of an object, for example, according to Whittaker, A translation is the operation changing the positions of all points of an object according to the formula → where is the same vector for each point of the object. When considering spacetime, a change of time coordinate is considered to be a translation, for example, the Galilean group and the Poincaré group include translations with respect to time
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Rotation (mathematics)
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Rotation in mathematics is a concept originating in geometry. Any rotation is a motion of a space that preserves at least one point. It can describe, for example, the motion of a body around a fixed point. A clockwise rotation is a negative magnitude so a counterclockwise turn has a positive magnitude, mathematically, a rotation is a map. All rotations about a fixed point form a group under composition called the rotation group, for example, in two dimensions rotating a body clockwise about a point keeping the axes fixed is equivalent to rotating the axes counterclockwise about the same point while the body is kept fixed. These two types of rotation are called active and passive transformations, the rotation group is a Lie group of rotations about a fixed point. This fixed point is called the center of rotation and is identified with the origin. The rotation group is a point stabilizer in a group of motions. For a particular rotation, The axis of rotation is a line of its fixed points and they exist only in n >2. The plane of rotation is a plane that is invariant under the rotation, unlike the axis, its points are not fixed themselves. The axis and the plane of a rotation are orthogonal, a representation of rotations is a particular formalism, either algebraic or geometric, used to parametrize a rotation map. This meaning is somehow inverse to the meaning in the group theory, rotations of spaces of points and of respective vector spaces are not always clearly distinguished. The former are sometimes referred to as affine rotations, whereas the latter are vector rotations, see the article below for details. A motion of a Euclidean space is the same as its isometry, but a rotation also has to preserve the orientation structure. The improper rotation term refers to isometries that reverse the orientation, in the language of group theory the distinction is expressed as direct vs indirect isometries in the Euclidean group, where the former comprise the identity component. Any direct Euclidean motion can be represented as a composition of a rotation about the fixed point, there are no non-trivial rotations in one dimension. In two dimensions, only a single angle is needed to specify a rotation about the origin – the angle of rotation that specifies an element of the circle group. The rotation is acting to rotate an object counterclockwise through an angle θ about the origin, composition of rotations sums their angles modulo 1 turn, which implies that all two-dimensional rotations about the same point commute