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Re: circumference of an ellipse
*To*: mathgroup at smc.vnet.net
*Subject*: [mg19292] Re: circumference of an ellipse
*From*: "Allan Hayes" <hay at haystack.demon.co.uk>
*Date*: Sat, 14 Aug 1999 01:45:16 -0400
*References*: <7p017c$778@smc.vnet.net>
*Sender*: owner-wri-mathgroup at wolfram.com
Marcel,
circumference[a_, b_] :=
Integrate[Sqrt[D[a Cos[t], t]^2 + D[b Sin[t], t]^2], {t, 0, 2Pi}]
Check:
circumference[1, 1]
2 Pi
N[circumference[2, 1], 30]
9.68844822054767619842850319639
N[circumference[5, 1], 30]
21.0100445396890009446991645885
N[circumference[100, 20], 30]
420.200890793780018893983291769
The above computes an exact symbolic value which is then approximatied to 30
places by N[--,30]
Immediate numerical integration may be sufficient for your needs and is
probably quicker.
Ncircumference[a_, b_] := NIntegrate[
Evaluate[Sqrt[D[a Cos[t], t]^2 + D[b Sin[t], t]^2]], {t, 0, 2Pi // N}]
Ncircumference[2, 1]
9.68845
Ncircumference[5, 1]
21.01
Ncircumference[100, 20]
420.201
If you use this a lot you might make an interpolating function.
Allan
---------------------
Allan Hayes
Mathematica Training and Consulting
Leicester UK
www.haystack.demon.co.uk
hay at haystack.demon.co.uk
Voice: +44 (0)116 271 4198
Fax: +44 (0)870 164 0565
M.J.M. Maes <mar.maes at wxs.nl> wrote in message
news:7p017c$778 at smc.vnet.net...
> I made a little spreadsheetfile to calculate the circumference (length) of
> an ellipse using an iterative process.
> When I compare the results with the results of a formula to approximate
the
> lenght
> - which is pi(3(a+b)-sqrt((a+3b)(3a+b))) - the difference is greater then
I
> expected (a few percents).
> This may be a shortcoming of the approximation formula, or of my
> worksheetformulae.
> Could someone please give me the exact results (in 10 or 15 digits) of
some
> examples?
> a=2; b=1
> a=5; b=1
> a=100; b=26
>
> Thanks in advance!
>
> Marcel
>
>
>
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