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Cesàro Equation
In geometry, the Cesàro equation of a plane curve is an equation relating the curvature () at a point of the curve to the arc length () from the start of the curve to the given point. It may also be given as an equation relating the radius of curvature () to arc length. (These are equivalent because .) Two congruent curves will have the same Cesàro equation. Cesàro equations are named after Ernesto Cesàro. Log-aesthetic curves The family of log-aesthetic curves is determined in the general (\alpha \ne 0) case by the following intrinsic equation: R(s)^\alpha = c_0s + c_1 This is equivalent to the following explicit formula for curvature: \kappa(s) = (c_0s+c_1)^ Further, the c_1 constant above represents simple re-parametrization of the arc length parameter, while c_0 is equivalent to uniform scaling, so log-aesthetic curves are fully characterized by the \alpha parameter. In the special case of \alpha=0, the log-aesthetic curve becomes Nielsen's spiral, with the following ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Geometry
Geometry (; ) is a branch of mathematics concerned with properties of space such as the distance, shape, size, and relative position of figures. Geometry is, along with arithmetic, one of the oldest branches of mathematics. A mathematician who works in the field of geometry is called a ''List of geometers, geometer''. Until the 19th century, geometry was almost exclusively devoted to Euclidean geometry, which includes the notions of point (geometry), point, line (geometry), line, plane (geometry), plane, distance, angle, surface (mathematics), surface, and curve, as fundamental concepts. Originally developed to model the physical world, geometry has applications in almost all sciences, and also in art, architecture, and other activities that are related to graphics. Geometry also has applications in areas of mathematics that are apparently unrelated. For example, methods of algebraic geometry are fundamental in Wiles's proof of Fermat's Last Theorem, Wiles's proof of Fermat's ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Plane Curve
In mathematics, a plane curve is a curve in a plane that may be 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 repr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Equation
In mathematics, an equation is a mathematical formula that expresses the equality of two expressions, by connecting them with the equals sign . The word ''equation'' and its cognates in other languages may have subtly different meanings; for example, in French an ''équation'' is defined as containing one or more variables, while in English, any well-formed formula consisting of two expressions related with an equals sign is an equation. Solving an equation containing variables consists of determining which values of the variables make the equality true. The variables for which the equation has to be solved are also called unknowns, and the values of the unknowns that satisfy the equality are called solutions of the equation. There are two kinds of equations: identities and conditional equations. An identity is true for all values of the variables. A conditional equation is only true for particular values of the variables. The " =" symbol, which appears in every equati ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Curvature
In mathematics, curvature is any of several strongly related concepts in geometry that intuitively measure the amount by which a curve deviates from being a straight line or by which a surface deviates from being a plane. If a curve or surface is contained in a larger space, curvature can be defined ''extrinsically'' relative to the ambient space. Curvature of Riemannian manifolds of dimension at least two can be defined ''intrinsically'' without reference to a larger space. For curves, the canonical example is that of a circle, which has a curvature equal to the reciprocal of its radius. Smaller circles bend more sharply, and hence have higher curvature. The curvature ''at a point'' of a differentiable curve is the curvature of its osculating circle — that is, the circle that best approximates the curve near this point. The curvature of a straight line is zero. In contrast to the tangent, which is a vector quantity, the curvature at a point is typically a scalar q ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Arc Length
Arc length is the distance between two points along a section of a curve. Development of a formulation of arc length suitable for applications to mathematics and the sciences is a problem in vector calculus and in differential geometry. In the most basic formulation of arc length for a vector valued curve (thought of as the trajectory of a particle), the arc length is obtained by integrating speed, the magnitude of the velocity vector over the curve with respect to time. Thus the length of a continuously differentiable curve (x(t),y(t)), for a\le t\le b, in the Euclidean plane is given as the integral L = \int_a^b \sqrt\,dt, (because \sqrt is the magnitude of the velocity vector (x'(t),y'(t)), i.e., the particle's speed). The defining integral of arc length does not always have a closed-form expression, and numerical integration may be used instead to obtain numerical values of arc length. Determining the length of an irregular arc segment by approximating the arc segment as ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Radius Of Curvature (mathematics)
In differential geometry, the radius of curvature, , is the reciprocal of the curvature. For a curve, it equals the radius of the circular arc which best approximates the curve at that point. For surfaces, the radius of curvature is the radius of a circle that best fits a normal section or combinations thereof. Definition In the case of a space curve, the radius of curvature is the length of the curvature vector. In the case of a plane curve, then is the absolute value of : R \equiv \left, \frac \ = \frac, where is the arc length from a fixed point on the curve, is the tangential angle and is the curvature. Formula In two dimensions If the curve is given in Cartesian coordinates as , i.e., as the graph of a function, then the radius of curvature is (assuming the curve is differentiable up to order 2) R =\left, \frac \\,, where y' = \frac\,, y'' = \frac, and denotes the absolute value of . If the curve is given parametrically by functions and , then the radiu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Congruence (geometry)
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. More formally, two sets of points are called congruent if, and only if, one can be transformed into the other by an isometry, i.e., a combination of rigid motions, namely a translation, a rotation, and a reflection. This means that either object can be repositioned and reflected (but not resized) so as to coincide precisely with the other object. Therefore, two distinct plane figures on a piece of paper are congruent if they can be cut out and then matched 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 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ernesto Cesàro
Ernesto Cesàro (12 March 1859 – 12 September 1906) was an Italian mathematician who worked in the field of differential geometry. He wrote a book, ''Lezioni di geometria intrinseca'' (Naples, 1890), on this topic, in which he also describes fractal, space-filling curves, partly covered by the larger class of de Rham curves, but are still known today in his honor as Cesàro curves. He is known also for his 'averaging' method for the 'Cesàro-summation' of divergent series, known as the Cesàro mean. Biography After a rather disappointing start of his academic career and a journey through Europe—with the most important stop at Liège, where his older brother Giuseppe Raimondo Pio Cesàro was teaching mineralogy at the local university—Ernesto Cesàro graduated from the University of Rome in 1887, while he was already part of the Royal Science Society of Belgium for the numerous works that he had already published. The following year, he obtained a mathematics chair at t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Nielsen's Spiral
In mathematics, trigonometric integrals are a family of nonelementary integrals involving trigonometric functions. Sine integral The different sine integral definitions are \operatorname(x) = \int_0^x\frac\,dt \operatorname(x) = -\int_x^\infty\frac\,dt~. Note that the integrand \frac is the sinc function, and also the zeroth spherical Bessel function. Since is an even entire function (holomorphic over the entire complex plane), is entire, odd, and the integral in its definition can be taken along any path connecting the endpoints. By definition, is the antiderivative of whose value is zero at , and is the antiderivative whose value is zero at . Their difference is given by the Dirichlet integral, \operatorname(x) - \operatorname(x) = \int_0^\infty\frac\,dt = \frac \quad \text \quad \operatorname(x) = \frac + \operatorname(x) ~. In signal processing, the oscillations of the sine integral cause overshoot and ringing artifacts when using the sinc filter, and fr ... [...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). This curve is also referred to as a clothoid or Cornu spiral.Levien, Raph"The Euler spiral: a mathematical history."Rapp. tech (2008). The behavior of Fresnel integrals can be illustrated by an Euler spiral, a connection first made by Marie Alfred Cornu in 1874. Euler's spiral is a type of superspiral that has the property of a monotonic curvature function. The Euler spiral has applications to diffraction computations. They are also widely used in railway engineering, railway and highway engineering to design transition curves between straight and curved sections of railways 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 be ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Logarithmic Spiral
A logarithmic spiral, equiangular spiral, or growth spiral is a self-similarity, self-similar spiral curve that often appears in nature. The first to describe a logarithmic spiral was Albrecht Dürer (1525) who called it an "eternal line" ("ewige Linie"). More than a century later, the curve was discussed by René Descartes, Descartes (1638), and later extensively investigated by Jacob Bernoulli, who called it ''Spira mirabilis'', "the marvelous spiral". The logarithmic spiral is distinct from the Archimedean spiral in that the distances between the turnings of a logarithmic spiral increase in a geometric progression, whereas for an Archimedean spiral these distances are constant. Definition In polar coordinates (r, \varphi) the logarithmic spiral can be written as r = ae^,\quad \varphi \in \R, or \varphi = \frac \ln \frac, with e (mathematical constant), e being the base of natural logarithms, and a > 0, k\ne 0 being real constants. In Cartesian coordinates The logarithmi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Involute
In mathematics, an involute (also known as an evolvent) is a particular type of curve that is dependent on another shape or curve. An involute of a curve is the Locus (mathematics), locus of a point on a piece of taut string as the string is either unwrapped from or wrapped around the curve. The evolute of an involute is the original curve. It is generalized by the Roulette (curve), roulette family of curves. That is, the involutes of a curve are the roulettes of the curve generated by a straight line. The notions of the involute and evolute of a curve were introduced by Christiaan Huygens in his work titled ''Horologium Oscillatorium, Horologium oscillatorium sive de motu pendulorum ad horologia aptato demonstrationes geometricae'' (1673), where he showed that the involute of a cycloid is still a cycloid, thus providing a method for constructing the cycloidal pendulum, which has the useful property that its period is independent of the amplitude of oscillation. Involute of a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
