How to find the arc length of an ellipse
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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
I will add a few more thoughts to the topic.
For an ellipseand 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 substitutiongives
,
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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