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-   -   How to find arc length of an ellipse (https://www.askmehelpdesk.com/showthread.php?t=494413)

  • Aug 3, 2010, 01:58 AM
    mufeel
    How to find arc length of an ellipse
    How to find the arc length of an ellipse
  • Aug 3, 2010, 05:38 AM
    galactus
    Quote:

    Originally Posted by mufeel View Post
    How to find the arc length of an ellipse

    This is difficult. This is why Elliptic Integrals were 'invented'.

    Do a search on this site. It seems to me I answered something like this in the past.

    See here for a nice paper on the topic:

    http://www.maa.org/editorial/euler/H...%20ellipse.pdf
  • Aug 4, 2010, 09:03 AM
    galactus
    I will add a few more thoughts to the topic.

    For an ellipse and the eccentricity, e, is

    The equation for the ellipse can be solved for y.
    , then we can differentiate and get .

    Square this and it in the arc length formula.



    The substitution gives
    ,

    This is the result for the arc length from x = 0 to x/a in the first

    quadrant, beginning at the point (0,b) on the y-axis. We have the whole

    arc for x/a = 1, and four times this is the circumference of the ellipse.

    This does not appear to be a particularly difficult integral, but all

    attempts to express it in terms of elementary functions fail. It must be

    considered as defining a new function, the elliptic integral of the second

    kind. Often we express it in a different form using the substitution , where t is called the amplitude, and k = e the modulus

    of the function E(t,k). In this case, .

    The whole arc corresponds to the upper limit , and , the complete

    elliptic integral of the second kind. We found an integral for the arc

    length, defined a new function, and expressed the arc length in terms of

    this new function. The reason for the name "elliptic".

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