Formula to find the length of a parabola line

[SolarFreak]

I have been been searching for a while before I desided to ask this. searching google and text books with not any good answers yet.

 

In ploting out the parabolas I have been using the formula

x^2 = 4py p = focal point

how do I calc the length of a parabolic line.

 

I did it the low tech way and graph'd it on paper then used a string to measure the lenght.

 

I do have an idea how to do it, you measure the distance from each point to point and sum the distances using

d = sqrt ( (x2 - x1)^2 + (y2 - y1)^2 )

 

is that right? is there a faster, better, cooler way to find the total distance?

 

Thanks

 

Tony

[UncleAl]

http://mathworld.wolfram.com/Parabola.html

"curvature, arc length, and tangential angle"

http://xanadu.math.utah.edu/java/ApproxLength.html

http://en.wikipedia.org/wiki/Arc_length

[SolarFreak]

thanks for the links Im reading them over now.

 

 

is this forumal right?

http://www.cs.mtu.edu/~shene/COURSES/cs201/NOTES/chap02/p-length.html

 

to test it I used data from a known parabola and its not even in the ballpark.

[Qfwfq]

I do have an idea how to do it, you measure the distance from each point to point and sum the distances using d = sqrt ( (x2 - x1)^2 + (y2 - y1)^2 )

This is the right idea, except that "point to point" is a bit of a mouthful, to be you need to perform an integral in a suitable manner. If you choose some number n of points along the line you can get an approximation that improves if you add more points. You need especially many points in the less straight parts of the curve. However a parabola is an algebric form so it is easy to work out the integral exactly.

 

[math]\int dx\;1 = x + c[/math]

 

[math]\int dx\;x = \frac{1}{2}x^2 + c[/math]

 

[math]\int dx\;x^2 = \frac{1}{3}x^3 + c[/math]

[CraigD]

is this forumal right?

http://www.cs.mtu.edu/~shene/COURSES/cs201/NOTES/chap02/p-length.html

It agrees with a quick approximation I made, except that it seems to give 2 times the actual arc length, assume the terms “base b” = x, and “height h” = y = [math]k x^2[/math] where [math]k[/math] is an constant, and the parabola passes through (0,0).

 

PS: The derivation of this formula doesn’t seem too hard (though I’ve not had a chance to get past the first steps).

[SolarFreak]

When I use that forumal on that page I get 177 his answer is 266

 

Height of a parabola :

100.0

Base of a parabola :

78.5

 

Height = 100.

Base = 78.5

Length = 266.149445

[CraigD]

Height = 100.

Base = 78.5

Length = 266.149445

That’s the same result I get. Note that this must be divided by 2 to give the length of a segment of a parabola between (0,0) and (78.5,100) of about 133.074727. The entire parabola should also pass through (-78.5,100).

 

So, Give your original formula x^2 =4py, p= x^2/(4y) = 15.405625.

 

The string-measuring technique you described should give a result close to 133.

 

You can perform a rough “reality check” on the obtained length value of about 133.074727 by considering that it must be greater than about 127.130838, the distance of a straight line (0,0)-(78.5,100), and 178.5, the sum of length of 2 straight lines (0,0)-(78.5,0) and (78.5,0)-(78.5,100).

[SolarFreak]

yeah when I setup this parabola in my calc

http://forum.teamfc3s.org/parabola/

and using the d = sqrt ( (x2 - x1)^2 + (y2 - y1)^2 ) for each point I plot on the parabola I get a length of 266.14 the length is at the bottom of the page.