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Perimeter of an ellipse

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An ellipse has two axes and two foci
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Unlike most other elementary shapes, such as the circle and square, there is no algebraic equation to determine the perimeter of an ellipse. Throughout history, a large number of equations for approximations and estimates have been made for the perimeter of an ellipse.

Mathematical background

An ellipse is defined by two axes: the major axis (the longest diameter, a {\displaystyle a} ) and the minor axis (the shortest diameter, b {\displaystyle b} ). The exact perimeter P {\displaystyle P} of an ellipse is given by the integral:

P = 4 a ∫ 0 π 2 1 − e 2 s i n 2 θ   d θ {\displaystyle P=4a\int _{0}^{\frac {\pi }{2}}{\sqrt {1-e^{2}sin^{2}\theta }}\ d\theta }

where e {\displaystyle e} is the eccentricity of the ellipse, defined as e = 1 − b 2 a 2 {\displaystyle e={\sqrt {1-{\frac {b^{2}}{a^{2}}}}}}

The integral used to find the area does not have a closed-form solution in terms of elementary functions. Another solution for the perimeter, this time using infinite sums, is:

P = 2 a π ( 1 − ∑ i = 1 ∞ 2 i ! 2 2 i × i ! × e 2 i 2 i − 1 ) {\displaystyle P=2a\pi (1-\sum _{i=1}^{\infty }{\frac {2i!^{2}}{2^{i}\times i!}}\times {\frac {e^{2i}}{2i-1}})}

Approximations

Because the exact computation involves elliptic integrals, several approximations have been developed over time.

Ramanujan's approximations

Indian mathematician Srinivasa Ramanujan proposed multiple approximations:

First Approximation:

P ≈ π ( 3 ( a + b ) − ( 3 a + b ) ( a + 3 b ) ) {\displaystyle P\approx \pi (3(a+b)-{\sqrt {(3a+b)(a+3b)}})}

Second Approximation:

P ≈ π ( a + b ) ( 1 + 3 h 10 + 4 − 3 h ) {\displaystyle P\approx \pi (a+b)(1+{\frac {3h}{10+{\sqrt {4-3h}}}})}

where h = ( a − b ) 2 ( a + b ) 2 {\displaystyle h={\frac {(a-b)^{2}}{(a+b)^{2}}}}

Simple arithmetic-geometric mean approximation

P ≈ 2 π a 2 + b 2 2 {\displaystyle P\approx 2\pi {\sqrt {\frac {a^{2}+b^{2}}{2}}}}

This formula is simpler than most perimeter formulas but less accurate for highly eccentric ellipses.

Approximations made from programs

In more recent years, computer programs have been used to find and calculate more precise approximations of the perimeter of an ellipse. In an online video about the perimeter of an ellipse, recreational mathematician and YouTuber Matt Parker, using a computer program, calculated numerous approximations for the perimeter of an ellipse. Approximations Parker found include:

P ≈ π ( 6 a 5 + 3 b 4 ) {\displaystyle P\approx \pi ({\frac {6a}{5}}+{\frac {3b}{4}})}

P ≈ π ( 53 a 3 + 717 b 35 − 269 a 2 + 667 a b + 371 b 2 ) {\displaystyle P\approx \pi ({\frac {53a}{3}}+{\frac {717b}{35}}-{\sqrt {269a^{2}+667ab+371b^{2}}})}

References

  1. Chandrupatla, Tirupathi; Osler, Thomas (2010). "The Perimeter of an Ellipse" (PDF). The Mathematical Scientist. 35 (2): 122–131.
  2. Abbott, Paul (2009). "On the Perimeter of an Ellipse" (PDF). The Mathematica Journal. 11 (2): 2. doi:10.3888/tmj.11.2-4.
  3. Villarino, Mark B. (February 1, 2008). "Ramanujan's Perimeter of an Ellipse". arXiv:math.CA/0506384.
  4. Stand-up Maths (2020-09-05). Why is there no equation for the perimeter of an ellipse‽. Retrieved 2024-12-16 – via YouTube.

See also

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