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  2. Arc length - Wikipedia

    en.wikipedia.org/wiki/Arc_length

    Arc length is the distance ... Since it is straightforward to calculate the length of ... The mapping that transforms from polar coordinates to rectangular ...

  3. Lemniscate of Bernoulli - Wikipedia

    en.wikipedia.org/wiki/Lemniscate_of_Bernoulli

    The determination of the arc length of arcs of the lemniscate leads to elliptic integrals, as was discovered in the eighteenth century. Around 1800, the elliptic functions inverting those integrals were studied by C. F. Gauss (largely unpublished at the time, but allusions in the notes to his Disquisitiones Arithmeticae ).

  4. Lemniscate elliptic functions - Wikipedia

    en.wikipedia.org/wiki/Lemniscate_elliptic_functions

    The lemniscate sine relates the arc length to the x coordinate in the rectangular elastica. The inverse lemniscate sine also describes the arc length s relative to the x coordinate of the rectangular elastica. [36] This curve has y coordinate and arc length:

  5. Polar coordinate system - Wikipedia

    en.wikipedia.org/wiki/Polar_coordinate_system

    The complex number z can be represented in rectangular form as = + where i is the imaginary unit, or can alternatively be written in polar form as = (⁡ + ⁡) and from there, by Euler's formula, [14] as = = ⁡. where e is Euler's number, and φ, expressed in radians, is the principal value of the complex number function arg applied to x + iy ...

  6. Spherical trigonometry - Wikipedia

    en.wikipedia.org/wiki/Spherical_trigonometry

    In practical applications it is often small: for example the triangles of geodetic survey typically have a spherical excess much less than 1' of arc. [14] On the Earth the excess of an equilateral triangle with sides 21.3 km (and area 393 km 2) is approximately 1 arc second. There are many formulae for the excess.

  7. Circular segment - Wikipedia

    en.wikipedia.org/wiki/Circular_segment

    The arc length, from the familiar geometry of a circle, is = The area a of the circular segment is equal to the area of the circular sector minus the area of the triangular portion (using the double angle formula to get an equation in terms of ):

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  9. Lemniscate - Wikipedia

    en.wikipedia.org/wiki/Lemniscate

    The lemniscate of Bernoulli and its two foci. In algebraic geometry, a lemniscate (/ l ɛ m ˈ n ɪ s k ɪ t / or / ˈ l ɛ m n ɪ s ˌ k eɪ t,-k ɪ t /) [1] is any of several figure-eight or ∞-shaped curves.