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Cornu Spiral
An Euler spiral is a curve whose curvature changes linearly with its curve length (the curvature of a circular curve is equal to the reciprocal of the radius). Euler spirals are also commonly referred to as spiros, clothoids, or Cornu spirals. Euler spirals have applications to diffraction computations. They are also widely used in railway and highway engineering to design transition curves between straight and curved sections of railway or roads. A similar application is also found in photonic integrated circuits. The principle of linear variation of the curvature of the transition curve between a tangent and a circular curve defines the geometry of the Euler spiral: *Its curvature begins with zero at the straight section (the tangent) and increases linearly with its curve length. *Where the Euler spiral meets the circular curve, its curvature becomes equal to that of the latter. Applications Track transition curve To travel along a circular path, an object needs to be ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Euler Spiral
An Euler spiral is a curve whose curvature changes linearly with its curve length (the curvature of a circular curve is equal to the reciprocal of the radius). Euler spirals are also commonly referred to as spiros, clothoids, or Cornu spirals. Euler spirals have applications to diffraction computations. They are also widely used in railway and highway engineering to design transition curves between straight and curved sections of railway or roads. A similar application is also found in photonic integrated circuits. The principle of linear variation of the curvature of the transition curve between a tangent and a circular curve defines the geometry of the Euler spiral: *Its curvature begins with zero at the straight section (the tangent) and increases linearly with its curve length. *Where the Euler spiral meets the circular curve, its curvature becomes equal to that of the latter. Applications Track transition curve To travel along a circular path, an object needs to be su ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Racing Line
In motorsport, the racing line is the optimal path around a race course. In most cases, the line makes use of the entire width of the track to lengthen the radius of a turn: entering at the outside edge, touching the "apex"—a point on the inside edge—then exiting the turn by returning outside. Description Driving the racing line is a primary technique for minimizing the overall course time. As the optimal path around a race course, the racing line can often be glimpsed on the asphalt in the form of tire skid marks. A.J. Baime described its formation in the early laps of a race at Le Mans: Racing line optimization A primary goal of the racing driver is to determine the optimum line around a race track. This optimum line may vary depending on whether a driver wishes to achieve a minimum lap time during a qualifying session, conserve tires and fuel, or fend off a pass from another driver during a race. Race tracks are often broken down into separate elements such as standar ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fresnel Integral
250px, Plots of and . The maximum of is about . If the integrands of and were defined using instead of , then the image would be scaled vertically and horizontally (see below). The Fresnel integrals and are two transcendental functions named after Augustin-Jean Fresnel that are used in optics and are closely related to the error function (). They arise in the description of near-field Fresnel diffraction phenomena and are defined through the following integral representations: S(x) = \int_0^x \sin\left(t^2\right)\,dt, \quad C(x) = \int_0^x \cos\left(t^2\right)\,dt. The simultaneous parametric plot of and is the Euler spiral (also known as the Cornu spiral or clothoid). Definition 250px, Fresnel integrals with arguments instead of converge to instead of . The Fresnel integrals admit the following power series expansions that converge for all : \begin S(x) &= \int_0^x \sin\left(t^2\right)\,dt = \sum_^(-1)^n \frac, \\ C(x) &= \int_0^x \cos\left(t^2\right)\,dt = ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Archimedean Spiral
The Archimedean spiral (also known as the arithmetic spiral) is a spiral named after the 3rd-century BC Greek mathematician Archimedes. It is the locus corresponding to the locations over time of a point moving away from a fixed point with a constant speed along a line that rotates with constant angular velocity. Equivalently, in polar coordinates it can be described by the equation r = a + b\cdot\theta with real numbers and . Changing the parameter moves the centerpoint of the spiral outward from the origin (positive toward and negative toward ) essentially through a rotation of the spiral, while controls the distance between loops. From the above equation, it can thus be stated: position of particle from point of start is proportional to angle as time elapses. Archimedes described such a spiral in his book '' On Spirals''. Conon of Samos was a friend of his and Pappus states that this spiral was discovered by Conon. Derivation of general equation of spiral A p ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Power Series
In mathematics, a power series (in one variable) is an infinite series of the form \sum_^\infty a_n \left(x - c\right)^n = a_0 + a_1 (x - c) + a_2 (x - c)^2 + \dots where ''an'' represents the coefficient of the ''n''th term and ''c'' is a constant. Power series are useful in mathematical analysis, where they arise as Taylor series of infinitely differentiable functions. In fact, Borel's theorem implies that every power series is the Taylor series of some smooth function. In many situations, ''c'' (the ''center'' of the series) is equal to zero, for instance when considering a Maclaurin series. In such cases, the power series takes the simpler form \sum_^\infty a_n x^n = a_0 + a_1 x + a_2 x^2 + \dots. Beyond their role in mathematical analysis, power series also occur in combinatorics as generating functions (a kind of formal power series) and in electronic engineering (under the name of the Z-transform). The familiar decimal notation for real numbers can also be viewed as an ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Numerical Stability
In the mathematical subfield of numerical analysis, numerical stability is a generally desirable property of numerical algorithms. The precise definition of stability depends on the context. One is numerical linear algebra and the other is algorithms for solving ordinary and partial differential equations by discrete approximation. In numerical linear algebra, the principal concern is instabilities caused by proximity to singularities of various kinds, such as very small or nearly colliding eigenvalues. On the other hand, in numerical algorithms for differential equations the concern is the growth of round-off errors and/or small fluctuations in initial data which might cause a large deviation of final answer from the exact solution. Some numerical algorithms may damp out the small fluctuations (errors) in the input data; others might magnify such errors. Calculations that can be proven not to magnify approximation errors are called ''numerically stable''. One of the common task ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Osculation
In differential geometry, an osculating curve is a plane curve from a given family that has the highest possible order of contact with another curve. That is, if ''F'' is a family of smooth curves, ''C'' is a smooth curve (not in general belonging to ''F''), and ''p'' is a point on ''C'', then an osculating curve from ''F'' at ''p'' is a curve from ''F'' that passes through ''p'' and has as many of its derivatives at ''p'' equal to the derivatives of ''C'' as possible... The term derives from the Latinate root "osculate", to kiss, because the two curves contact one another in a more intimate way than simple tangency. Examples Examples of osculating curves of different orders include: *The tangent line to a curve ''C'' at a point ''p'', the osculating curve from the family of straight lines. The tangent line shares its first derivative (slope) with ''C'' and therefore has first-order contact with ''C''.. *The osculating circle to ''C'' at ''p'', the osculating curve from the famil ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Easement Curve
A track transition curve, or spiral easement, is a mathematically-calculated curve on a section of highway, or railroad track, in which a straight section changes into a curve. It is designed to prevent sudden changes in lateral (or centripetal) acceleration. In plane (viewed from above), the start of the transition of the horizontal curve is at infinite radius, and at the end of the transition, it has the same radius as the curve itself and so forms a very broad spiral. At the same time, in the vertical plane, the outside of the curve is gradually raised until the correct degree of bank is reached. If such an easement were not applied, the lateral acceleration of a rail vehicle would change abruptly at one point (the tangent point where the straight track meets the curve) with undesirable results. With a road vehicle, a transition curve allows the driver to alter the steering in a gradual manner. History On early railroads, because of the low speeds and wide-radius curves emp ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Whiskers
Vibrissae (; singular: vibrissa; ), more generally called Whiskers, are a type of stiff, functional hair used by mammals to touch, sense their environment. These hairs are finely specialised for this purpose, whereas other types of hair are coarser as tactile sensors. Although whiskers are specifically those found around the face, vibrissae are known to grow in clusters at various places around the body. Most mammals have them, including all non-human primates and especially Nocturnality, nocturnal mammals. Whiskers are sensitive tactile hairs that aid navigation, locomotion, exploration, hunting, social touch and perform other functions. This article is primarily about the specialised sensing hairs of mammals, but some birds, fish, insects, crustaceans and other arthropods are known to have similar structures also used to sense the environment. Etymology Vibrissae (from Latin 'to vibrate') from the characteristic motion seen in a small rodent that is otherwise sitting sti ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
YouTube
YouTube is a global online video platform, online video sharing and social media, social media platform headquartered in San Bruno, California. It was launched on February 14, 2005, by Steve Chen, Chad Hurley, and Jawed Karim. It is owned by Google, and is the List of most visited websites, second most visited website, after Google Search. YouTube has more than 2.5 billion monthly users who collectively watch more than one billion hours of videos each day. , videos were being uploaded at a rate of more than 500 hours of content per minute. In October 2006, YouTube was bought by Google for $1.65 billion. Google's ownership of YouTube expanded the site's business model, expanding from generating revenue from advertisements alone, to offering paid content such as movies and exclusive content produced by YouTube. It also offers YouTube Premium, a paid subscription option for watching content without ads. YouTube also approved creators to participate in Google's Google AdSens ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Map Projection
In cartography, map projection is the term used to describe a broad set of transformations employed to represent the two-dimensional curved surface of a globe on a plane. In a map projection, coordinates, often expressed as latitude and longitude, of locations from the surface of the globe are transformed to coordinates on a plane. Projection is a necessary step in creating a two-dimensional map and is one of the essential elements of cartography. All projections of a sphere on a plane necessarily distort the surface in some way and to some extent. Depending on the purpose of the map, some distortions are acceptable and others are not; therefore, different map projections exist in order to preserve some properties of the sphere-like body at the expense of other properties. The study of map projections is primarily about the characterization of their distortions. There is no limit to the number of possible map projections. More generally, projections are considered in several fi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Globe
A globe is a spherical model of Earth, of some other celestial body, or of the celestial sphere. Globes serve purposes similar to maps, but unlike maps, they do not distort the surface that they portray except to scale it down. A model globe of Earth is called a terrestrial globe. A model globe of the celestial sphere is called a ''celestial globe''. A globe shows details of its subject. A terrestrial globe shows landmasses and water bodies. It might show nations and major cities and the network of latitude and longitude lines. Some have raised relief to show mountains and other large landforms. A celestial globe shows notable stars, and may also show positions of other prominent astronomical objects. Typically, it will also divide the celestial sphere into constellations. The word ''globe'' comes from the Latin word ''globus'', meaning "sphere". Globes have a long history. The first known mention of a globe is from Strabo, describing the Globe of Crates from about 150&nb ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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