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  2. Arc length - Wikipedia

    en.wikipedia.org/wiki/Arc_length

    There are continuous curves on which every arc (other than a single-point arc) has infinite length. An example of such a curve is the Koch curve. Another example of a curve with infinite length is the graph of the function defined by f(x) = x sin(1/x) for any open set with 0 as one of its delimiters and f(0) = 0.

  3. Line element - Wikipedia

    en.wikipedia.org/wiki/Line_element

    The coordinate-independent definition of the square of the line element ds in an n-dimensional Riemannian or Pseudo Riemannian manifold (in physics usually a Lorentzian manifold) is the "square of the length" of an infinitesimal displacement [2] (in pseudo Riemannian manifolds possibly negative) whose square root should be used for computing curve length: = = (,) where g is the metric tensor ...

  4. Rayleigh–Ritz method - Wikipedia

    en.wikipedia.org/wiki/Rayleigh–Ritz_method

    Hence M = [m 1, m 2] and K = [k 1, k 2]. A mode shape is assumed for the system, with two terms, one of which is weighted by a factor B , e.g. Y = [1, 1] + B [1, −1]. Simple harmonic motion theory says that the velocity at the time when deflection is zero, is the angular frequency ω {\displaystyle \omega } times the deflection (y) at time of ...

  5. Radius of curvature - Wikipedia

    en.wikipedia.org/wiki/Radius_of_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. [1] [2] [3]

  6. Euler spiral - Wikipedia

    en.wikipedia.org/wiki/Euler_spiral

    A double-end Euler spiral. The curve continues to converge to the points marked, as t tends to positive or negative infinity. 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.

  7. Tractrix - Wikipedia

    en.wikipedia.org/wiki/Tractrix

    The arc length of one branch between x = x 1 and x = x 2 is a ln ⁠ y 1 / y 2 ⁠. The area between the tractrix and its asymptote is ⁠ π a 2 / 2 ⁠, which can be found using integration or Mamikon's theorem. The envelope of the normals of the tractrix (that is, the evolute of the tractrix) is the catenary (or chain curve) given by y = a ...

  8. Sagitta (geometry) - Wikipedia

    en.wikipedia.org/wiki/Sagitta_(geometry)

    In the following equations, denotes the sagitta (the depth or height of the arc), equals the radius of the circle, and the length of the chord spanning the base of the arc. As 1 2 l {\displaystyle {\tfrac {1}{2}}l} and r − s {\displaystyle r-s} are two sides of a right triangle with r {\displaystyle r} as the hypotenuse , the Pythagorean ...

  9. Euler–Bernoulli beam theory - Wikipedia

    en.wikipedia.org/wiki/Euler–Bernoulli_beam_theory

    The curve () describes the deflection of the beam in the direction at some position (recall that the beam is modeled as a one-dimensional object). q {\displaystyle q} is a distributed load, in other words a force per unit length (analogous to pressure being a force per area); it may be a function of x {\displaystyle x} , w {\displaystyle w ...