How to determine the arc length of ellipse? I want to

Lorraine Harvey 2021-12-16 Answered
How to determine the arc length of ellipse?
I want to determine the length of an arc from the ellipse in the picture below:
\(\displaystyle\theta={50}^{{\circ}}\)
\(\displaystyle{a}={3.05}{c}{m}\)
\(\displaystyle{b}={2.23}{c}{m}\)
\(\displaystyle{d}=?{c}{m}\)
How can I determine the length of d?

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Expert Answer

Melissa Moore
Answered 2021-12-17 Author has 4742 answers
Let \(\displaystyle{a}={3.05},{b}={2.23}\). Then a parametric equation for the ellipse is \(\displaystyle{x}={a}{\cos{{t}}},{y}={b}{\sin{}}\). When \(\displaystyle{t}={0}\) the point is at \(\displaystyle{\left({a},{0}\right)}={\left({3.05},{0}\right)}\), the starting point of the arc on the ellipse whose length you seek. Now it's important to realize that the parameter t is not the central angle, so you need to get the value of t which corresponds to the top end of your arc. At that end you have \(\displaystyle{\frac{{{y}}}{{{x}}}}={\tan{{50}}}\) (degrees). And in terms of t you have \(\displaystyle{\frac{{{y}}}{{{x}}}}={\left({\frac{{{b}}}{{{a}}}}\right)}{\tan{}}\). Solving for t then gives
\(\displaystyle{t}={t}_{{{1}}}={a}{r}{c}{\tan{{\left({\frac{{{a}}}{{{b}}}}{\tan{{50}}}\right)}}}\).
[note I'd suggest using radians here, replacing the 50 by \(\displaystyle{\frac{{{5}\pi}}{{{18}}}}\).]
For the arclength use the general formula of integrating \(\displaystyle\sqrt{{{x}'^{{{2}}}+{y}'^{{{2}}}}}\) for t in the desired range. In your case \(\displaystyle{x}′=-{a}{\sin{{t}}},{y}′={b}{\cos{}}\), so that you are integrating
\(\displaystyle\sqrt{{{a}^{{{2}}}{{\sin}^{{{2}}}{t}}+{b}^{{{2}}}{{\cos}^{{{2}}}{t}}}}\)
with respect to t from 0 to the above \(\displaystyle{t}_{{{1}}}\). There not being a simple closed form for the antiderivative (it's an "elliptic integral), the simplest approach now would be to do the integral numerically. This seems the more appropriate in your problem as you only know a,b to two decimals, apparently.
* When I did this numerically on maple I got about 2.531419 for the arclength.
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Jenny Bolton
Answered 2021-12-18 Author has 6337 answers
You can compute this as
\(\displaystyle{d}={b}{E}{\left({{\tan}^{{-{1}}}{\left({\frac{{{a}}}{{{b}}}}{\tan{{\left(\theta\right)}}}\right)}}{\mid}{1}-{\left({\frac{{{a}}}{{{b}}}}\right)}^{{{2}}}\right)}\)
using the incomplete elliptic integral of the second kind \(\displaystyle{E}{\left(\psi{\mid}{m}\right)}\). In Mathematica-Syntax (and suitable for Wolfram Alpha) this can be written as
\(\displaystyle{2.23}\cdot{E}{l}{l}{i}{p}{t}{i}{c}\ {E}{\left[{A}{r}{c}{\tan{{\left[{\frac{{{3.05}}}{{{2.23}}}}\cdot{\tan{{\left[{50}^{{\circ}}\right]}}}\right]}}},{1}-{\left({\frac{{{3.05}}}{{{2.23}}}}\right)}^{{{2}}}\right]}\)
I adapted this from this post which investigates the converse problem (given arc length, find angle) but along the way treats this direction of the problem as well. As noted there, this angle conversion will only work for the first and last quadrant. Otherwise, either adjust the angle or look at that post for an alternative formula to use in its place.
With a few more digits of precision, the answer is returned as 2.5314195265536624417 which essentially matches both the other answers here. Of course, printing that many digits in the answer is very bad style if the input is only given to two decimals. It does show that the numerical integration by Jyrki is a bit less precise than what coffeemath did, but even he should theoretically have rounded in the other direction.
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nick1337
Answered 2021-12-28 Author has 10701 answers

Giving a Mathematica calculation. Same result as coffeemath (+1)
\(In[1]:= Arc \tan [3.05 \cdot \tan[\frac{5 \pi}{18}]/2.23]\)
\(Out[1]= 1.02051\)
\(In[2]:= x=3.05\ \cos[t];\)
\(In[3]:= y=2.23 \sin[t];\)
\(KIn[4]:= N\ Integrate[\sqrt{D(x,t)^{2}+D(y,t)^{2}},\{t,0,1.02051\}]\)
\(Out[4]= 2.53143\)

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