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Radiodrome
In geometry, a radiodrome is the pursuit curve followed by a point that is pursuing another linearly-moving point. The term is derived from the Greek words and . The classic (and best-known) form of a radiodrome is known as the "dog curve"; this is the path a dog follows when it swims across a stream with a current after something it has spotted on the other side. Because the dog drifts with the current, it will have to change its heading; it will also have to swim further than if it had taken the optimal heading. This case was described by Pierre Bouguer in 1732. A radiodrome may alternatively be described as the path a dog follows when chasing a hare, assuming that the hare runs in a straight line at a constant velocity. Mathematical analysis Introduce a coordinate system with origin at the position of the dog at time zero and with ''y''-axis in the direction the hare is running with the constant speed . The position of the hare at time zero is with and at time it is T ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pursuit Curve
In geometry, a curve of pursuit is a curve constructed by analogy to having a point or points representing pursuers and pursuees; the curve of pursuit is the curve traced by the pursuers. With the paths of the pursuer and pursuee parameterized in time, the pursuee is always on the pursuer's tangent. That is, given , the pursuer (follower), and , the pursued (leader), for every with there is an such that :L(t) = F(t) + xF'\!(t). History The pursuit curve was first studied by Pierre Bouguer in 1732. In an article on navigation, Bouguer defined a curve of pursuit to explore the way in which one ship might maneuver while pursuing another. Leonardo da Vinci has occasionally been credited with first exploring curves of pursuit. However Paul J. Nahin, having traced such accounts as far back as the late 19th century, indicates that these anecdotes are unfounded. Single pursuer The path followed by a single pursuer, following a pursuee that moves at constant speed on a line, is ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pierre Bouguer
Pierre Bouguer () (16 February 1698, Croisic – 15 August 1758, Paris) was a French mathematician, geophysicist, geodesist, and astronomer. He is also known as "the father of naval architecture". Career Bouguer's father, Jean Bouguer, one of the best hydrographers of his time, was Regius Professor of hydrography at Le Croisic in lower Brittany, and author of a treatise on navigation. He taught his sons Pierre and Jan at their home, where he also taught private students. In 1714, at the age of 16, Pierre was appointed to succeed his deceased father as professor of hydrography. In 1727 he gained the prize given by the French Academy of Sciences for his paper ''On the masting of ships'', beating Leonhard Euler; and two other prizes, one for his dissertation ''On the best method of observing the altitude of stars at sea'', the other for his paper ''On the best method of observing the variation of the compass at sea''. These were published in the Prix de l'Académie des Science ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Geometry
Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is called a ''geometer''. Until the 19th century, geometry was almost exclusively devoted to Euclidean geometry, which includes the notions of point, line, plane, distance, angle, surface, and curve, as fundamental concepts. During the 19th century several discoveries enlarged dramatically the scope of geometry. One of the oldest such discoveries is Carl Friedrich Gauss' ("remarkable theorem") that asserts roughly that the Gaussian curvature of a surface is independent from any specific embedding in a Euclidean space. This implies that surfaces can be studied ''intrinsically'', that is, as stand-alone spaces, and has been expanded into the theory of manifolds and Riemannian geometry. Later in the 19th century, it appeared that geometries ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Greek Language
Greek ( el, label=Modern Greek, Ελληνικά, Elliniká, ; grc, Ἑλληνική, Hellēnikḗ) is an independent branch of the Indo-European family of languages, native to Greece, Cyprus, southern Italy (Calabria and Salento), southern Albania, and other regions of the Balkans, the Black Sea coast, Asia Minor, and the Eastern Mediterranean. It has the longest documented history of any Indo-European language, spanning at least 3,400 years of written records. Its writing system is the Greek alphabet, which has been used for approximately 2,800 years; previously, Greek was recorded in writing systems such as Linear B and the Cypriot syllabary. 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. The Greek language holds a very important place in the history of the Western world. Beginning with the epics of Homer, ancient Greek literature includes many works of lasting impo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dog Curve
In geometry, a radiodrome is the pursuit curve followed by a point that is pursuing another linearly-moving point. The term is derived from the Greek words and . The classic (and best-known) form of a radiodrome is known as the "dog curve"; this is the path a dog follows when it swims across a stream with a current after something it has spotted on the other side. Because the dog drifts with the current, it will have to change its heading; it will also have to swim further than if it had taken the optimal heading. This case was described by Pierre Bouguer in 1732. A radiodrome may alternatively be described as the path a dog follows when chasing a hare, assuming that the hare runs in a straight line at a constant velocity. Mathematical analysis Introduce a coordinate system with origin at the position of the dog at time zero and with ''y''-axis in the direction the hare is running with the constant speed . The position of the hare at time zero is with and at time it is T ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mice Problem
In mathematics, the mice problem is a continuous pursuit–evasion problem in which a number of mice (or insects, dogs, missiles, etc.) are considered to be placed at the corners of a regular polygon. In the classic setup, each then begins to move towards its immediate neighbour (clockwise or anticlockwise). The goal is often to find out at what time the mice meet. The most common version has the mice starting at the corners of a unit square, moving at unit speed. In this case they meet after a time of one unit, because the distance between two neighboring mice always decreases at a speed of one unit. More generally, for a regular polygon of n unit-length sides, the distance between neighboring mice decreases at a speed of 1 - \cos(2\pi/n), so they meet after a time of 1/\bigl(1 - \cos(2\pi/n)\bigr). Path of the mice For all regular polygons, each mouse traces out a pursuit curve in the shape of a logarithmic spiral A logarithmic spiral, equiangular spiral, or growth spir ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Plane Curves
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 (including piecewise smooth plane curves), and algebraic plane curves. Plane curves also include the Jordan curves (curves that enclose a region of the plane but need not be smooth) and the graphs of continuous functions. Symbolic representation A plane curve can often be represented in Cartesian coordinates by an implicit equation of the form f(x,y)=0 for some specific function ''f''. If this equation can be solved explicitly for ''y'' or ''x'' – that is, rewritten as y=g(x) or x=h(y) for specific function ''g'' or ''h'' – then this provides an alternative, explicit, form of the representation. A plane curve can also often be represented in Cartesian coordinates by a parametric equation of the form (x,y)=(x(t), y(t)) for specific functions x(t) and y(t). Plane curves can sometimes also be ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Differential Equations
In mathematics, a differential equation is an equation that relates one or more unknown functions and their derivatives. In applications, the functions generally represent physical quantities, the derivatives represent their rates of change, and the differential equation defines a relationship between the two. Such relations are common; therefore, differential equations play a prominent role in many disciplines including engineering, physics, economics, and biology. Mainly the study of differential equations consists of the study of their solutions (the set of functions that satisfy each equation), and of the properties of their solutions. Only the simplest differential equations are solvable by explicit formulas; however, many properties of solutions of a given differential equation may be determined without computing them exactly. Often when a closed-form expression for the solutions is not available, solutions may be approximated numerically using computers. The theory of d ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Analytic Geometry
In classical mathematics, analytic geometry, also known as coordinate geometry or Cartesian geometry, is the study of geometry using a coordinate system. This contrasts with synthetic geometry. Analytic geometry is used in physics and engineering, and also in aviation, Aerospace engineering, rocketry, space science, and spaceflight. It is the foundation of most modern fields of geometry, including Algebraic geometry, algebraic, Differential geometry, differential, Discrete geometry, discrete and computational geometry. Usually the Cartesian coordinate system is applied to manipulate equations for planes, straight lines, and circles, often in two and sometimes three dimensions. Geometrically, one studies the Euclidean plane (two dimensions) and Euclidean space. As taught in school books, analytic geometry can be explained more simply: it is concerned with defining and representing geometric shapes in a numerical way and extracting numerical information from shapes' numerical defin ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
