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Stadium (geometry)
A stadium is a two-dimensional geometric shape constructed of a rectangle with semicircles at a pair of opposite sides. The same shape is known also as a pill shape, discorectangle, obround, or sausage body. The shape is based on a stadium, a place used for athletics and horse racing tracks. A stadium may be constructed as the Minkowski sum of a disk and a line segment. Alternatively, it is the neighborhood of points within a given distance from a line segment. A stadium is a type of oval. However, unlike some other ovals such as the ellipses, it is not an algebraic curve because different parts of its boundary are defined by different equations. Formulas The perimeter of a stadium is calculated by the formula P = 2 (\pi r+a) where ''a'' is the length of the straight sides and ''r'' is the radius of the semicircles. With the same parameters, the area of the stadium is A = \pi r^2 + 2ra = r(\pi r + 2a). Bunimovich stadium When this shape is used in the study of dynamical ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Stadium Geometry
A stadium (: stadiums or stadia) is a place or venue for (mostly) outdoor sports, concerts, or other events and consists of a field or stage completely or partially surrounded by a tiered structure designed to allow spectators to stand or sit and view the event. Pausanias of Athens, Pausanias noted that for about half a century the only event at the ancient Greek Olympic festival was the race that comprised one length of the Stadion (unit), stadion at Olympia, Greece, Olympia, where the word "stadium" originated. Most of the stadiums with a capacity of at least 10,000 are used for association football. Other popular stadium sports include gridiron football, baseball, cricket, the various codes of Rugby football, rugby, field lacrosse, bandy, and bullfighting. Many large sports venues are also used for concerts. Etymology "Stadium" is the Latin form of the Greek word "Stadion (unit), stadion" (''στάδιον''), a measure of length equalling the length of 600 human feet. A ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Oval
An oval () is a closed curve in a plane which resembles the outline of an egg. The term is not very specific, but in some areas of mathematics (projective geometry, technical drawing, etc.), it is given a more precise definition, which may include either one or two axes of symmetry of an ellipse. In common English, the term is used in a broader sense: any shape which reminds one of an egg. The three-dimensional version of an oval is called an ovoid. Oval in geometry The term oval when used to describe curves in geometry is not well-defined, except in the context of projective geometry. Many distinct curves are commonly called ovals or are said to have an "oval shape". Generally, to be called an oval, a plane curve should ''resemble'' the outline of an egg or an ellipse. In particular, these are common traits of ovals: * they are differentiable (smooth-looking), simple (not self-intersecting), convex, closed, plane curves; * their shape does not depart much from that of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Line Of Symmetry
In mathematics, reflection symmetry, line symmetry, mirror symmetry, or mirror-image symmetry is symmetry with respect to a reflection. That is, a figure which does not change upon undergoing a reflection has reflectional symmetry. In two-dimensional space, there is a line/axis of symmetry, in three-dimensional space, there is a plane of symmetry. An object or figure which is indistinguishable from its transformed image is called mirror symmetric. Symmetric function In formal terms, a mathematical object is symmetric with respect to a given operation such as reflection, rotation, or translation, if, when applied to the object, this operation preserves some property of the object. The set of operations that preserve a given property of the object form a group. Two objects are symmetric to each other with respect to a given group of operations if one is obtained from the other by some of the operations (and vice versa). The symmetric function of a two-dimensional figure ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Capsule (geometry)
A capsule (from Latin '' capsula'', "small box or chest"), or stadium of revolution, is a basic three-dimensional geometric shape consisting of a cylinder with hemispherical ends. Another name for this shape is spherocylinder. It can also be referred to as an oval although the sides (either vertical or horizontal) are straight parallel. Usages The shape is used for some objects like containers for pressurised gases, building domes, and pharmaceutical capsules. In chemistry and physics, this shape is used as a basic model for non-spherical particles. It appears, in particular as a model for the molecules in liquid crystals or for the particles in granular matter. Formulas The volume V of a capsule is calculated by adding the volume of a ball of radius r (that accounts for the two hemispheres) to the volume of the cylindrical part. Hence, if the cylinder has height h, :V = \frac\pi r^3 + (\pi r^2h)= \pi r^2 \left (\fracr + h \right ). The surface area of a capsule of radi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lyapunov Exponent
In mathematics, the Lyapunov exponent or Lyapunov characteristic exponent of a dynamical system is a quantity that characterizes the rate of separation of infinitesimally close trajectory, trajectories. Quantitatively, two trajectories in phase space with initial separation vector \boldsymbol_0 diverge (provided that the divergence can be treated within the linearized approximation) at a rate given by , \boldsymbol(t) , \approx e^ , \boldsymbol_0 , where \lambda is the Lyapunov exponent. The rate of separation can be different for different orientations of initial separation vector. Thus, there is a spectrum of Lyapunov exponents—equal in number to the dimensionality of the phase space. It is common to refer to the largest one as the maximal Lyapunov exponent (MLE), because it determines a notion of predictability for a dynamical system. A positive MLE is usually taken as an indication that the system is chaos theory, chaotic (provided some other conditions are met, e.g., ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Chaos Theory
Chaos theory is an interdisciplinary area of Scientific method, scientific study and branch of mathematics. It focuses on underlying patterns and Deterministic system, deterministic Scientific law, laws of dynamical systems that are highly sensitive to initial conditions. These were once thought to have completely random states of disorder and irregularities. Chaos theory states that within the apparent randomness of chaotic complex systems, there are underlying patterns, interconnection, constant feedback loops, repetition, self-similarity, fractals and self-organization. The butterfly effect, an underlying principle of chaos, describes how a small change in one state of a deterministic nonlinear system can result in large differences in a later state (meaning there is sensitive dependence on initial conditions). A metaphor for this behavior is that a butterfly flapping its wings in Brazil can cause or prevent a tornado in Texas. Text was copied from this source, which is avai ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Leonid Bunimovich
Leonid Abramowich Bunimovich (born August 1, 1947) is a Soviet and American mathematician, who made fundamental contributions to the theory of Dynamical Systems, Statistical Physics and various applications. Bunimovich received his bachelor's degree in 1967, master's degree in 1969 and PhD in 1973 from the University of Moscow. His masters and PhD thesis advisor was Yakov G. Sinai. In 1986 (after Perestroika started) he finally received Doctor of Sciences degree in "Theoretical and Mathematical Physics". Bunimovich is a Regents' Professor of Mathematics at the Georgia Institute of Technology. In 1990-91 Bunimovich was Volkswagen Professor of Physics in the Bielefeld University, he is a Fellow of the Institute of Physics and awarded Humboldt Prize in Physics. Biography His Master's proved that some classes of quadratic maps of an interval have aabsolutely continuous invariant measureand strong stochastic properties. Bunimovich is mostly known for discovery of a fundamental mecha ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bunimovich Stadium
A dynamical billiard is a dynamical system in which a particle alternates between free motion (typically as a straight line) and specular reflections from a boundary. When the particle hits the boundary it reflects from it without loss of speed (i.e. elastic collisions). Billiards are Hamiltonian idealizations of the game of billiards, but where the region contained by the boundary can have shapes other than rectangular and even be multidimensional. Dynamical billiards may also be studied on non-Euclidean geometries; indeed, the first studies of billiards established their ergodic motion on surfaces of constant negative curvature. The study of billiards which are kept out of a region, rather than being kept in a region, is known as outer billiard theory. The motion of the particle in the billiard is a straight line, with constant energy, between reflections with the boundary (a geodesic if the Riemannian metric of the billiard table is not flat). All reflections are specular: ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Area (geometry)
Area is the measure of a region's size on a surface. The area of a plane region or ''plane area'' refers to the area of a shape or planar lamina, while ''surface area'' refers to the area of an open surface or the boundary of a three-dimensional object. Area can be understood as the amount of material with a given thickness that would be necessary to fashion a model of the shape, or the amount of paint necessary to cover the surface with a single coat. It is the two-dimensional analogue of the length of a curve (a one-dimensional concept) or the volume of a solid (a three-dimensional concept). Two different regions may have the same area (as in squaring the circle); by synecdoche, "area" sometimes is used to refer to the region, as in a "polygonal area". The area of a shape can be measured by comparing the shape to squares of a fixed size. In the International System of Units (SI), the standard unit of area is the square metre (written as m2), which is the area of a square ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Perimeter
A perimeter is the length of a closed boundary that encompasses, surrounds, or outlines either a two-dimensional shape or a one-dimensional line. The perimeter of a circle or an ellipse is called its circumference. Calculating the perimeter has several practical applications. A calculated perimeter is the length of fence required to surround a yard or garden. The perimeter of a wheel/circle (its circumference) describes how far it will roll in one revolution. Similarly, the amount of string wound around a spool is related to the spool's perimeter; if the length of the string was exact, it would equal the perimeter. Formulas The perimeter is the distance around a shape. Perimeters for more general shapes can be calculated, as any path, with \int_0^L \mathrms, where L is the length of the path and ds is an infinitesimal line element. Both of these must be replaced by algebraic forms in order to be practically calculated. If the perimeter is given as a closed piecewise ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Algebraic Curve
In mathematics, an affine algebraic plane curve is the zero set of a polynomial in two variables. A projective algebraic plane curve is the zero set in a projective plane of a homogeneous polynomial in three variables. An affine algebraic plane curve can be completed in a projective algebraic plane curve by homogenization of a polynomial, homogenizing its defining polynomial. Conversely, a projective algebraic plane curve of homogeneous equation can be restricted to the affine algebraic plane curve of equation . These two operations are each inverse function, inverse to the other; therefore, the phrase algebraic plane curve is often used without specifying explicitly whether it is the affine or the projective case that is considered. If the defining polynomial of a plane algebraic curve is irreducible polynomial, irreducible, then one has an ''irreducible plane algebraic curve''. Otherwise, the algebraic curve is the union of one or several irreducible curves, called its ''Irreduc ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
