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  2. Arc length - Wikipedia

    en.wikipedia.org/wiki/Arc_length

    For example, consider the problem of finding the length of a quarter of the unit circle by numerically integrating the arc length integral. The upper half of the unit circle can be parameterized as y = 1 − x 2 . {\displaystyle y={\sqrt {1-x^{2}}}.}

  3. Calculus of variations - Wikipedia

    en.wikipedia.org/wiki/Calculus_of_Variations

    The calculus of variations may be said to begin with Newton's minimal resistance problem in 1687, followed by the brachistochrone curve problem raised by Johann Bernoulli (1696). [2] It immediately occupied the attention of Jacob Bernoulli and the Marquis de l'Hôpital, but Leonhard Euler first elaborated the subject, beginning in 1733.

  4. Differentiable curve - Wikipedia

    en.wikipedia.org/wiki/Differentiable_curve

    Differential geometry takes another path: curves are represented in a parametrized form, and their geometric properties and various quantities associated with them, such as the curvature and the arc length, are expressed via derivatives and integrals using vector calculus.

  5. Line element - Wikipedia

    en.wikipedia.org/wiki/Line_element

    Since is an arbitrary "square of the arc length", completely defines the metric, and it is therefore usually best to consider the expression for as a definition of the metric tensor itself, written in a suggestive but non tensorial notation: = This identification of the square of arc length with the metric is even more easy to see in n-dimensional general curvilinear coordinates q = (q 1, q 2 ...

  6. Elliptic integral - Wikipedia

    en.wikipedia.org/wiki/Elliptic_integral

    In integral calculus, an elliptic integral is one of a number of related functions defined as the value of certain integrals, which were first studied by Giulio Fagnano and Leonhard Euler (c. 1750). Their name originates from their originally arising in connection with the problem of finding the arc length of an ellipse.

  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. Intrinsic equation - Wikipedia

    en.wikipedia.org/wiki/Intrinsic_equation

    The Cesàro equation is obtained as a relation between arc length and curvature. The equation of a circle (including a line) for example is given by the equation κ ( s ) = 1 r {\displaystyle \kappa (s)={\tfrac {1}{r}}} where s {\displaystyle s} is the arc length, κ {\displaystyle \kappa } the curvature and r {\displaystyle r} the radius of ...

  9. Area of a circle - Wikipedia

    en.wikipedia.org/wiki/Area_of_a_circle

    Let the length of A′B be c n, which we call the complement of s n; thus c n 2 +s n 2 = (2r) 2. Let C bisect the arc from A to B, and let C′ be the point opposite C on the circle. Thus the length of CA is s 2n, the length of C′A is c 2n, and C′CA is itself a right triangle on diameter C′C.

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