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Arc length s of a logarithmic spiral as a function of its parameter θ. Arc length is the distance between two points along a section of a curve. Development of a formulation of arc length suitable for applications to mathematics and the sciences is a focus of calculus.
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 Around 1800, the elliptic functions inverting those integrals were studied by C. F. Gauss (largely unpublished at the time, but allusions in the notes to his Disquisitiones Arithmeticae ).
Equivalently, in polar coordinates (r, θ) it can be described by the equation = with real number b. Changing the parameter b controls the distance between loops. From the above equation, it can thus be stated: position of the particle from point of start is proportional to angle θ as time elapses.
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 ):
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: y = ∫ x 1 t 2 d t 1 − t 4 , s = arcsl x = ∫ 0 x d t 1 − t 4 {\displaystyle y=\int _{x}^{1}{\frac {t^{2}\mathop {\mathrm {d} t} }{\sqrt {1-t^{4}}}},\quad s ...
Logarithmic spiral (pitch 10°) A section of the Mandelbrot set following a logarithmic spiralA logarithmic spiral, equiangular spiral, or growth spiral is a self-similar spiral curve that often appears in nature.
The arc length of the positive branch of the Fermat's spiral from the origin can also be defined by hypergeometric functions 2 F 1 (a, b; c; z) and the incomplete beta function B(z; a, b): [4]
12.3 Arc length. 12.4 Curvature. ... the equation of a standard ellipse centered at the origin with width ... is the center of the rectangle ,,, ...