|
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 2D 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 \, \qquad\mbox\quad y' = \frac,\quad y'' = \frac, and denotes the absolute value of . If the curve is given parametrically by functions and , then the rad ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Radius Of Curvature
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 2D 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 \, \qquad\mbox\quad y' = \frac,\quad 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] |
Parametric Equation
In mathematics, a parametric equation defines a group of quantities as functions of one or more independent variables called parameters. Parametric equations are commonly used to express the coordinates of the points that make up a geometric object such as a curve or surface, in which case the equations are collectively called a parametric representation or parameterization (alternatively spelled as parametrisation) of the object. For example, the equations :\begin x &= \cos t \\ y &= \sin t \end form a parametric representation of the unit circle, where ''t'' is the parameter: A point (''x'', ''y'') is on the unit circle if and only if there is a value of ''t'' such that these two equations generate that point. Sometimes the parametric equations for the individual scalar output variables are combined into a single parametric equation in vectors: :(x, y)=(\cos t, \sin t). Parametric representations are generally nonunique (see the "Examples in two dimensions" section b ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
AFM Probe
Atomic force microscopy (AFM) or scanning force microscopy (SFM) is a very-high-resolution type of scanning probe microscopy (SPM), with demonstrated resolution on the order of fractions of a nanometer, more than 1000 times better than the optical diffraction limit. Overview Atomic force microscopy (AFM) is a type of scanning probe microscopy (SPM), with demonstrated resolution on the order of fractions of a nanometer, more than 1000 times better than the optical diffraction limit. The information is gathered by "feeling" or "touching" the surface with a mechanical probe. Piezoelectric elements that facilitate tiny but accurate and precise movements on (electronic) command enable precise scanning. Despite the name, the Atomic Force Microscope does not use the Nuclear force. Abilities The AFM has three major abilities: force measurement, topographic imaging, and manipulation. In force measurement, AFMs can be used to measure the forces between the probe and the sample as a f ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Minimum Railway Curve Radius
The minimum railway curve radius is the shortest allowable design radius for the centerline of railway tracks under a particular set of conditions. It has an important bearing on construction costs and operating costs and, in combination with superelevation (difference in elevation of the two rails) in the case of train tracks, determines the maximum safe speed of a curve. The minimum radius of a curve is one parameter in the design of railway vehicles as well as trams; monorails and automated guideways are also subject to a minimum radius. History The first proper railway was the Liverpool and Manchester Railway, which opened in 1830. Like the tram roads that had preceded it over a hundred years, the L&M had gentle curves and gradients. Reasons for these gentle curves include the lack of strength of the track, which might have overturned if the curves were too sharp causing derailments. The gentler the curves, the greater the visibility, thus boosting safety via incre ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Printed Electronics
Printed electronics is a set of printing methods used to create electrical devices on various substrates. Printing typically uses common printing equipment suitable for defining patterns on material, such as screen printing, flexography, gravure, offset lithography, and inkjet. By electronic-industry standards, these are low-cost processes. Electrically functional electronic or optical inks are deposited on the substrate, creating active or passive devices, such as thin film transistors; capacitors; coils; resistors. Some researchers expect printed electronics to facilitate widespread, very low-cost, low-performance electronics for applications such as flexible displays, smart labels, decorative and animated posters, and active clothing that do not require high performance. Coatanéa, E., Kantola, V., Kulovesi, J., Lahti, L., Lin, R., & Zavodchikova, M. (2009). Printed Electronics, Now and Future. In Neuvo, Y., & Ylönen, S. (eds.), Bit Bang – Rays to the Future. Helsinki U ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Radius Of Curvature (optics)
Radius of curvature (ROC) has specific meaning and sign convention in optical design. A spherical lens or mirror surface has a center of curvature located either along or decentered from the system local optical axis. The vertex of the lens surface is located on the local optical axis. The distance from the vertex to the center of curvature is the radius of curvature of the surface. The sign convention for the optical radius of curvature is as follows: * If the vertex lies to the left of the center of curvature, the radius of curvature is positive. * If the vertex lies to the right of the center of curvature, the radius of curvature is negative. Thus when viewing a biconvex lens from the side, the left surface radius of curvature is positive, and the right radius of curvature is negative. Note however that ''in areas of optics other than design'', other sign conventions are sometimes used. In particular, many undergraduate physics textbooks use the Gaussian sign conve ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Beam (structure)
A beam is a structural element that primarily resists loads applied laterally to the beam's axis (an element designed to carry primarily axial load would be a strut or column). Its mode of deflection is primarily by bending. The loads applied to the beam result in reaction forces at the beam's support points. The total effect of all the forces acting on the beam is to produce shear forces and bending moments within the beams, that in turn induce internal stresses, strains and deflections of the beam. Beams are characterized by their manner of support, profile (shape of cross-section), equilibrium conditions, length, and their material. Beams are traditionally descriptions of building or civil engineering structural elements, where the beams are horizontal and carry vertical loads. However, any structure may contain beams, for instance automobile frames, aircraft components, machine frames, and other mechanical or structural systems. In these structures, any structural elem ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Arc Measurement
Arc measurement, sometimes degree measurement (german: Gradmessung), is the astrogeodetic technique of determining of the radius of Earth – more specifically, the local Earth radius of curvature of the figure of the Earth – by relating the latitude difference (sometimes also the longitude difference) and the geographic distance (arc length) surveyed between two locations on Earth's surface. The most common variant involves only astronomical latitudes and the meridian arc length and is called ''meridian arc measurement''; other variants may involve only astronomical longitude (''parallel arc measurement'') or both geographic coordinates (''oblique arc measurement''). Arc measurement campaigns in Europe were the precursors to the International Association of Geodesy (IAG). History The first known arc measurement was performed by Eratosthenes (240 BC) between Alexandria and Syene in what is now Egypt, determining the radius of the Earth with remarkable correctness. In the e ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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. Examples Some curves have a particularly simple representation by a Cesàro equation. Some examples are: * Line: \kappa = 0. * Circle: \kappa = \frac, where is the radius. * Logarithmic spiral: \kappa=\frac, where is a constant. * Circle involute: \kappa=\frac, where is a constant. * Cornu spiral: \kappa=Cs, where is a constant. * Catenary: \kappa=\frac. Related parameterizations The Cesàro equation of a curve is related to its Whewell equation The Whewell equation of a plane curve is an equation that relates the tangential angle () with arclength (), where the tang ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Vertex (curve)
In the geometry of plane curves, a vertex is a point of where the first derivative of curvature is zero. This is typically a local maximum or minimum of curvature, and some authors define a vertex to be more specifically a local extremum of curvature. However, other special cases may occur, for instance when the second derivative is also zero, or when the curvature is constant. For space curves, on the other hand, a vertex is a point where the torsion vanishes. Examples A hyperbola has two vertices, one on each branch; they are the closest of any two points lying on opposite branches of the hyperbola, and they lie on the principal axis. On a parabola, the sole vertex lies on the axis of symmetry and in a quadratic of the form: :ax^2 + bx + c\,\! it can be found by completing the square or by differentiation., p. 127. On an ellipse, two of the four vertices lie on the major axis and two lie on the minor axis. For a circle, which has constant curvature, every point is a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ellipse
In mathematics, an ellipse is a plane curve surrounding two focal points, such that for all points on the curve, the sum of the two distances to the focal points is a constant. It generalizes a circle, which is the special type of ellipse in which the two focal points are the same. The elongation of an ellipse is measured by its eccentricity e, a number ranging from e = 0 (the limiting case of a circle) to e = 1 (the limiting case of infinite elongation, no longer an ellipse but a parabola). An ellipse has a simple algebraic solution for its area, but only approximations for its perimeter (also known as circumference), for which integration is required to obtain an exact solution. Analytically, the equation of a standard ellipse centered at the origin with width 2a and height 2b is: : \frac+\frac = 1 . Assuming a \ge b, the foci are (\pm c, 0) for c = \sqrt. The standard parametric equation is: : (x,y) = (a\cos(t),b\sin(t)) \quad \text \quad 0\leq t\leq 2\pi. Ellipses ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Circle
A circle is a shape consisting of all points in a plane that are at a given distance from a given point, the centre. Equivalently, it is the curve traced out by a point that moves in a plane so that its distance from a given point is constant. The distance between any point of the circle and the centre is called the radius. Usually, the radius is required to be a positive number. A circle with r=0 (a single point) is a degenerate case. This article is about circles in Euclidean geometry, and, in particular, the Euclidean plane, except where otherwise noted. Specifically, a circle is a simple closed curve that divides the plane into two regions: an interior and an exterior. In everyday use, the term "circle" may be used interchangeably to refer to either the boundary of the figure, or to the whole figure including its interior; in strict technical usage, the circle is only the boundary and the whole figure is called a '' disc''. A circle may also be defined as a specia ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
