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Advanced Math/Section of an ellipse


Hi Janet,

Please don't feel obligated to answer this until after the holidays! It's just been bugging me and I wanted to ask while I had the time to sit down at a computer.

I'm trying to find the length of a certain section of an ellipse. I'm imagining a satellite in a polar elliptical orbit around the Earth, and I'm trying to figure out how much of its orbit would be in shadow, assuming the Earth's shadow stretches straight behind it, instead of in a cone. When I draw it out it looks like this: an ellipse with a circle around one of the foci (this circle is well within the ellipse) and two straight lines drawn parallel to the minor axis that are tangent to the circle and meet the ellipse (just on one side of the major axis). My question is how to find the length of the section of the ellipse that is bound by these two lines.

Thanks so much for your help!

Dear Andrew,

I'm afraid that you lost me at "polar elliptical orbit"!

This is how I visualize your description:
and you want to find the length of the black arc.

If this correct?

Dear Andrew,

I came up with an equation for the arc length, but it contains an integral that I could not solve. I will post the equations as updates to this page, but that may not happen before Monday (I'm out of town with no access to equation software).

As an interim measure, I have submitted the problem to the Question Pool in case one of the other volunteers knows how to solve it.

Dear Andrew,

Here's a website that shows the same steps I followed, up to the start of "eccentricity":

I posted your question on another board and got one response:
"Even Ramanujan had trouble with this (he came up with an approximation that's pretty good, but not perfect).  Short answer is this: There is no general formula for the perimeter of an ellipse.
Elliptic Integrals are a special kind of evil."

I have to concede defeat on this one. At least I learned some new stuff!

Derivative of ellipse:
Equation for length of arc:  

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Janet Yang


I can answer questions in Algebra, Basic Math, Calculus, Differential Equations, Geometry, Number Theory, and Word Problems. I would not feel comfortable answering questions in Probability and Statistics or Topology because I have not studied these in depth.


I tutor students (fifth through twelfth grades) and am a Top Contributor on Yahoo!Answers with over 24,000 math solutions.

Co-author of An Outline of Scientific Writing: For Researchers With English as a Foreign Language

I have a Bachelor's degree in Applied Mathematics from the University of California at Berkeley.

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