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Radius of curvature. Radius of curvature and center of curvature. In differential geometry, the radius of curvature, R, 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 ...
Euler–Bernoulli beam theory (also known as engineer's beam theory or classical beam theory) [1] is a simplification of the linear theory of elasticity which provides a means of calculating the load-carrying and deflection characteristics of beams. It covers the case corresponding to small deflections of a beam that is subjected to lateral ...
If k = −1, then (loosely speaking) one can say that i · a is the radius of curvature of the universe. a is the scale factor which is taken to be 1 at the present time. k is the current spatial curvature (when a = 1). If the shape of the universe is hyperspherical and R t is the radius of curvature (R 0 at the present), then a = R t / R 0
The Frenet–Serret formulas are frequently introduced in courses on multivariable calculus as a companion to the study of space curves such as the helix. A helix can be characterized by the height 2π h and radius r of a single turn. The curvature and torsion of a helix (with constant radius) are given by the formulas.
Dubins proved his result in 1957. In 1974 Harold H. Johnson proved Dubins' result by applying Pontryagin's maximum principle. [4] In particular, Harold H. Johnson presented necessary and sufficient conditions for a plane curve, which has bounded piecewise continuous curvature and prescribed initial and terminal points and directions, to have minimal length.
The curvature is the reciprocal of radius of curvature. That is, the curvature is =, where R is the radius of curvature [5] (the whole circle has this curvature, it can be read as turn 2π over the length 2π R). This definition is difficult to manipulate and to express in formulas.
The Gaussian function has a 1/e 2 diameter (2w as used in the text) about 1.7 times the FWHM.. At a position z along the beam (measured from the focus), the spot size parameter w is given by a hyperbolic relation: [1] = + (), where [1] = is called the Rayleigh range as further discussed below, and is the refractive index of the medium.
v. t. e. In physics, circular motion is a movement of an object along the circumference of a circle or rotation along a circular arc. It can be uniform, with a constant rate of rotation and constant tangential speed, or non-uniform with a changing rate of rotation. The rotation around a fixed axis of a three-dimensional body involves the ...