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Homework Help: Solving for arc length of an ellipse

  1. Feb 24, 2010 #1
    1. The problem statement, all variables and given/known data
    The task is to solve for the arc length of an ellipse numerically. a & b are given for an ellipse centered at the origin and a value for x is given.

    2. Relevant equations

    Equation of ellipse is given to be
    [tex]x^{2}/a^{2} + y^{2}/b^{2} = 1[/tex]
    and the equation to solve for the arc length is given as
    [tex]a \int^{\theta}_{0}\sqrt{1-k^{2} sin^{2}t} dt[/tex]
    Assuming a is the major axis

    3. The attempt at a solution
    The additional condition is that
    [tex]-a \leq x\leq a[/tex], so [tex]\theta[/tex] can be [tex]\ge 0.5 \pi[/tex]
    Since [tex]\int^{\theta}_{0}\sqrt{1-k^{2} sin^{2}t} dt[/tex] does not seem to work when [tex]{\theta} \ge 0.5 \pi[/tex]

    The only solution I can think of is as follows
    1. If x < 0
    2. pb4 = quarter the perimeter of the ellipse
    3. Set x= -x (reflect about the x axis)
    4. ptemp = arc length for the positive x
    5. The final answer would be p=pb4+(pb4-ptemp)

    This would mean evaluating equation [tex]\int^{\theta}_{0}\sqrt{1-k^{2} sin^{2}t} dt[/tex] twice.
    Is there a better solution to this problem?


    - Sid
  2. jcsd
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