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Alhazen's Problem
Alhazen's problem is a mathematical problem in optics concerning reflection in a spherical mirror. It asks for the point in the mirror where one given point reflects to another. The special case of a concave spherical mirror is also known as Alhazen's billiard problem, as it can be formulated equivalently as constructing a reflected path from one billiard ball to another on a circular billiard table. Other equivalent formulations ask for the shortest path from one point to the other that touches the circle, or for an ellipse that is tangent to the circle and has the given points as its foci. Although special cases of this problem were studied by Ptolemy, it is named for the 11th-century Arab mathematician Alhazen (''Ibn al-Haytham''), who formulated it more generally and presented a solution in his ''Book of Optics''. It has no straightedge and compass construction; instead, al-Haytham and others including Christiaan Huygens found solutions involving the intersection of coni ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Leonardo Da Vinci
Leonardo di ser Piero da Vinci (15 April 1452 - 2 May 1519) was an Italian polymath of the High Renaissance who was active as a painter, draughtsman, engineer, scientist, theorist, sculptor, and architect. While his fame initially rested on his achievements as a painter, he has also become known for #Journals and notes, his notebooks, in which he made drawings and notes on a variety of subjects, including anatomy, astronomy, botany, cartography, painting, and palaeontology. Leonardo is widely regarded to have been a genius who epitomised the Renaissance humanism, Renaissance humanist ideal, and his List of works by Leonardo da Vinci, collective works comprise a contribution to later generations of artists matched only by that of his younger contemporary Michelangelo. Born out of wedlock to a successful notary and a lower-class woman in, or near, Vinci, Tuscany, Vinci, he was educated in Florence by the Italian painter and sculptor Andrea del Verrocchio. He began his career ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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General Position
In algebraic geometry and computational geometry, general position is a notion of genericity for a set of points, or other geometric objects. It means the ''general case'' situation, as opposed to some more special or coincidental cases that are possible, which is referred to as special position. Its precise meaning differs in different settings. For example, generically, two lines in the plane intersect in a single point (they are not parallel or coincident). One also says "two generic lines intersect in a point", which is formalized by the notion of a ''generic point''. Similarly, three generic points in the plane are not collinear; if three points are collinear (even stronger, if two coincide), this is a degenerate case. This notion is important in mathematics and its applications, because degenerate cases may require an exceptional treatment; for example, when stating general theorems or giving precise statements thereof, and when writing computer programs (see '' generic ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Isosceles Triangle
In geometry, an isosceles triangle () is a triangle that has two Edge (geometry), sides of equal length and two angles of equal measure. Sometimes it is specified as having ''exactly'' two sides of equal length, and sometimes as having ''at least'' two sides of equal length, the latter version thus including the equilateral triangle as a special case. Examples of isosceles triangles include the isosceles right triangle, the Golden triangle (mathematics), golden triangle, and the faces of bipyramids and certain Catalan solids. The mathematical study of isosceles triangles dates back to ancient Egyptian mathematics and Babylonian mathematics. Isosceles triangles have been used as decoration from even earlier times, and appear frequently in architecture and design, for instance in the pediments and gables of buildings. The two equal sides are called the ''legs'' and the third side is called the base (geometry), ''base'' of the triangle. The other dimensions of the triangle, such ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Lewis Carroll
Charles Lutwidge Dodgson (27 January 1832 – 14 January 1898), better known by his pen name Lewis Carroll, was an English author, poet, mathematician, photographer and reluctant Anglicanism, Anglican deacon. His most notable works are ''Alice's Adventures in Wonderland'' (1865) and its sequel ''Through the Looking-Glass'' (1871). He was noted for his facility with word play, logic, and fantasy. His poems ''Jabberwocky'' (1871) and ''The Hunting of the Snark'' (1876) are classified in the genre of literary nonsense. Some of Alice's nonsensical wonderland logic reflects his published work on mathematical logic. Carroll came from a family of high-church Anglicanism, Anglicans, and pursued his clerical training at Christ Church, Oxford, where he lived for most of his life as a scholar, teacher and (necessarily for his academic fellowship at the time) Anglican deacon. Alice Liddell – a daughter of Henry Liddell, the Dean of Christ Church, Oxford, Dean of Christ Church – is wide ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Billiards
Cue sports are a wide variety of games of skill played with a cue stick, which is used to strike billiard balls and thereby cause them to move around a cloth-covered table bounded by elastic bumpers known as . Cue sports, a category of stick sports, may collectively be referred to as billiards, though this term has more specific connotations in some English dialects. There are three major subdivisions of games within cue sports: * Carom billiards, played on tables without , typically ten feet in length, including straight rail, balkline, one-cushion carom, three-cushion billiards, artistic billiards, and four-ball * Pocket billiards (or pool), played on six-pocket tables of seven, eight, nine, or ten-foot length, including among others eight-ball (the world's most widely played cue sport), nine-ball (the dominant professional game), ten-ball, straight pool (the formerly dominant pro game), one-pocket, and bank pool *Snooker, English billiards, and Russian pyra ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Ray (optics)
In optics, a ray is an idealized geometrical model of light or other electromagnetic radiation, obtained by choosing a curve that is perpendicular to the ''wavefronts'' of the actual light, and that points in the direction of energy flow. Rays are used to model the propagation of light through an optical system, by dividing the real light field up into discrete rays that can be computationally propagated through the system by the techniques of '' ray tracing''. This allows even very complex optical systems to be analyzed mathematically or simulated by computer. Ray tracing uses approximate solutions to Maxwell's equations that are valid as long as the light waves propagate through and around objects whose dimensions are much greater than the light's wavelength. '' Ray optics'' or ''geometrical optics'' does not describe phenomena such as diffraction, which require wave optics theory. Some wave phenomena such as interference can be modeled in limited circumstances by adding ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Catoptrics
Catoptrics (from ''katoptrikós'', "specular", from ''katoptron'' "mirror") deals with the phenomena of reflected light and image-forming optical systems using mirrors. A catoptric system is also called a ''catopter'' (''catoptre''). History Ancient Texts ''Catoptrics'' is the title of two texts from ancient Greece: *The Pseudo-Euclidean ''Catoptrics''. This book is attributed to Euclid, although the contents are a mixture of work dating from Euclid's time together with work which dates to the Roman period., accessed 31 January 2013 It has been argued that the book may have been compiled by the 4th century mathematician Theon of Alexandria. The book covers the mathematical theory of mirrors, particularly the images formed by plane and spherical concave mirrors. *Hero's ''Catoptrics''. Written by Hero of Alexandria, this work concerns the practical application of mirrors for visual effects. In the Middle Ages, this work was falsely ascribed to Ptolemy. It only survives in a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Geometrical Optics
Geometrical optics, or ray optics, is a model of optics that describes light Wave propagation, propagation in terms of ''ray (optics), rays''. The ray in geometrical optics is an abstract object, abstraction useful for approximating the paths along which light propagates under certain circumstances. The simplifying assumptions of geometrical optics include that light rays: * propagate in straight-line paths as they travel in a Homogeneity (physics), homogeneous medium * bend, and in particular circumstances may split in two, at the Interface (matter), interface between two dissimilar optical medium, media * follow curved paths in a medium in which the refractive index changes * may be absorbed or reflected. Geometrical optics does not account for certain optical effects such as diffraction and Interference (wave propagation), interference, which are considered in physical optics. This simplification is useful in practice; it is an excellent approximation when the wavelength is smal ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Specular Reflection
Specular reflection, or regular reflection, is the mirror-like reflection (physics), reflection of waves, such as light, from a surface. The law of reflection states that a reflected ray (optics), ray of light emerges from the reflecting surface at the same angle to the surface normal as the incident ray, but on the opposing side of the surface normal in the plane formed by the incident and reflected rays. The earliest known description of this behavior was recorded by Hero of Alexandria (Anno Domini, AD c. 10–70). Later, Ibn al-Haytham, Alhazen gave a complete statement of the law of reflection. He was first to state that the incident ray, the reflected ray, and the normal to the surface all lie in a same plane perpendicular to reflecting plane. Specular reflection may be contrasted with diffuse reflection, in which light is scattered away from the surface in a range of directions. Law of reflection When light encounters a boundary of a material, it is affected by the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Surface Normal
In geometry, a normal is an object (e.g. a line, ray, or vector) that is perpendicular to a given object. For example, the normal line to a plane curve at a given point is the infinite straight line perpendicular to the tangent line to the curve at the point. A normal vector is a vector perpendicular to a given object at a particular point. A normal vector of length one is called a unit normal vector or normal direction. A curvature vector is a normal vector whose length is the curvature of the object. Multiplying a normal vector by results in the opposite vector, which may be used for indicating sides (e.g., interior or exterior). In three-dimensional space, a surface normal, or simply normal, to a surface at point is a vector perpendicular to the tangent plane of the surface at . The vector field of normal directions to a surface is known as '' Gauss map''. The word "normal" is also used as an adjective: a line ''normal'' to a plane, the ''normal'' component of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |