Analytically and numerically computed arc length
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Hi,
I'm trying to compute the length of a curve defined in parametric form:
t = linspace(0,pi); % Actually t could go from 0 to any angle lower than 2*pi
r = 1 ./ ( 1 - t / (2*pi) );
x = r.*cos(t); y = r.*sin(t);
dx = diff(x); dy = diff(y);
l = sum( sqrt(dx.^2 + dy.^2) ); % Arc length. Linear aprox.
This way the length is equal to 4.4725.
If I do the calculations analytically, I find the length is:
l = -2*pi*log( 1 - angle/(2*pi) ); % Being the initial point angle = 0
using angle = pi the result is 4.3552.
What's the reason of this difference?
Thanks in advance.
采纳的回答
Teja Muppirala
2012-7-10
I'm not sure how you are analytically calculating path length. It seems from your expression that you just integrated r(t) straight up, which is not the correct way to do it. You need to use the formula for path length.
For example, it is given for cylindrical coordinates here:
http://tutorial.math.lamar.edu/Classes/CalcII/PolarArcLength.aspx (Google: arc length cylindrical coordinates)
That equation is difficult to integrate by hand, but it can be done symbolically in MATLAB very easily.
syms r t
r = 1 ./ ( 1 - t / (2*pi) );
ds_dt = sqrt(r^2 + diff(r)^2);
path_length = int(ds_dt,0,pi)
subs(path_length)
You get a long analytical expression for path length, which turns out to be the same answer as you got numerically, 4.47.
2 个评论
Bilgah Johnson
2020-12-11
i am using this code but I'm not able to get the answer. Please help as soon as possible.
Bilgah Johnson
2020-12-11
PLEASE HELP ME> HOW TO SOLVE THE ABOVE QUESTION USING THE CODE.
更多回答（1 个）
Carlos
2012-7-10
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1 个评论
Walter Roberson
2012-7-10
No, it's okay, it will help other people in future.
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