Consider an ellipse described by the equation $ \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 $. Is there a closed form expression for the midpoint of the arc which goes from $ (a,0) $ to $ (0,b) $ ?
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Midpoint in the sense of dividing the arclength in half? – John Hughes Jul 25 '17 at 21:11
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no, there is not – Will Jagy Jul 25 '17 at 21:11
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1Isn't it $1/8$ of the Circumference? – gammatester Jul 25 '17 at 21:14
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On the other hand, there IS an explicit formula for finding the point $(x', y')$ that is twice as far along the ellipse as some point $(x, y)$, a kind of elliptical double-angle formula, due to Fagnano. See https://books.google.com/books?id=oxH2CAAAQBAJ&pg=PA181&lpg=PA181&dq=doubling+arc+length+on+an+ellipse&source=bl&ots=Ea2LrLT2f9&sig=sRplsNYyYrz7FbM8B4DSgrXNHpA&hl=en&sa=X&ved=0ahUKEwjV9qG6rKXVAhWixlQKHVxiBl0Q6AEITTAG#v=onepage&q=doubling%20arc%20length%20on%20an%20ellipse&f=false – John Hughes Jul 25 '17 at 21:15
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@John: yes, in that sense. – MCL Jul 25 '17 at 21:21
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@MVL It involves finding $\phi$ from second elliptic integral $ E(e^2, \phi) $ – Narasimham Jul 25 '17 at 21:21
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Related to Calculating equidistant points around an ellipse arc, for the special case of 8 equidistant points, including the vertices. Linked posts there might be useful reading, too. – MvG Jul 26 '17 at 23:06