View Full Version : Arc length on a helix
ebbe
Feb 3, 2009, 05:15 AM
Please give the formula for calculating the arc length on a helix
galactus
Feb 3, 2009, 06:23 AM
Represent the helix by x=a\cdot cos(t), \;\ y=a\cdot sin(t), \;\ z=ct
'a' is the radius and c is the length in one turn divided by 2Pi.
The formula is L=2\pi\sqrt{a^{2}+c^{2}}
EXAMPLE:
Suppose we have a string wrapped around a cylinder of radius 12 inches. The length that will make one complete turn in, say, 20 inches is given by:
2\pi\sqrt{36+(\frac{10}{\pi})^{2}}=42.68 \;\ inches
Scourge
Mar 22, 2009, 07:28 PM
Represent the helix by x=a\cdot cos(t), \;\ y=a\cdot sin(t), \;\ z=ct
'a' is the radius and c is the length in one turn divided by 2Pi.
The formula is L=2\pi\sqrt{a^{2}+c^{2}}
EXAMPLE:
Suppose we have a string wrapped around a cylinder of radius 12 inches. The length that will make one complete turn in, say, 20 inches is given by:
2\pi\sqrt{36+(\frac{10}{\pi})^{2}}=42.68 \;\ inches
Thank you for that equation galactus - I waded through pages of calculus treatments of this question before finding your elegant solution.
But you made an error in your calculation when you squared half of the radius, rather than the whole radius. The correct answer is:
2\pi\sqrt{144+(\frac{10}{\pi})^{2}}=78.00571860464 764 \;\ inches
Since the length of a circle with a radius of 12 inches is
2\pi\ r=75.39822\;\ inches
the length of a helix of the same radius will go up from that number as the length of the helix increases.
galactus
Mar 23, 2009, 08:42 AM
I made a mistake. That should have said 12 inch diameter, not radius.
What I have is correct. It is a typo with the word radius. Thanks for the catch after all this time.