# Thread: Finding the length of a one turn spiral

1. How do I find the length of this spiral for ONE TURN ONLY The circle in the center is 7 inches in diameter, and the spiral moves outward at a constant rate. So how do I find its length?

I know of several websites that allow one to plug in a spiral's dimensions and get its length, but could you could explain the concept as well. I want to understand the theory behind it.

Thankyou

2.

3. Someone give your thoughts on this idea... the spiral could be represented by . If the diameter's we'd have coefficients of on the and . I don't know if this is a specific or general case but I guess we could throw a on the term because I'm guessing that's what "the spiral moves outward at a constant rate" means. And then we can use the formula for arc length and integrate from to . Would that work?

4. The picture I posted is too small to see. When I said it moves outward at a constant rate, it would be one sixteenth inches away from the central cirlce at a quarter turn, one eight inch away at a half turn, three sixteenths away at 3/4 turn and it make a 1/4 inch gap at the instant it completes the turn. Hope that made sense.

5. ok, I get it. That's different then. I wonder if my method would work for a regular helix though... Sorry I couldn't help.

6. yeh i think regular helixes have the form sin(t) + cos (t) + z(t)

7. Originally Posted by SteveC

How do I find the length of this spiral for ONE TURN ONLY The circle in the center is 7 inches in diameter, and the spiral moves outward at a constant rate. So how do I find its length?

I know of several websites that allow one to plug in a spiral's dimensions and get its length, but could you could explain the concept as well. I want to understand the theory behind it.

Thankyou
Believe this or not, a spring does not really change in length much until you get past 45 degrees vertically.

In other words if you have a round flat coil one rotation, and you keep it under 45 degrees vertically, the length will remain about the same as you open the spring up to about 45 degrees. After that it will tighten the circle.

I make spiral staircases so I know this from actually pulling a round tube that was rolled into a single loop, rolled the diameter of the staircase it was going on as a hand rail. When I pulled one side of the loop straight up, it just stretched and fit onto the staircase.

A flat handrail believe it or not, has to be rolled much larger then the diameter of the staircase. As you pull up the flat handrail it reduces in diameter. Very weird stuff.

Sincerely,

William McCormick

8. Believe this or not, a spring does not really change in length much until you get past 45 degrees vertically.

In other words if you have a round flat coil one rotation, and you keep it under 45 degrees vertically, the length will remain about the same as you open the spring up to about 45 degrees. After that it will tighten the circle.
I just wrapped 4 turns around a rubber band at maybe a 10 degree tilt, then I upped it to 45 degrees and it was very close to the same. That is somewhat strange. Once it gets past 45 degrees, it starts making extreme changes to the number of turns. It would be interesting to see a graph of the angle of inclination versus the loss of rotations. I bet the graph would not be linear at all, because from 0 to 10 degrees we get basically no change, and from 80 to 90 degrees we lose all the rotations completely.

If you were to draw a rectangle, connect the diagonal corners and rolled it into a cylinder, the diagonal would form a spiral. If you then took a piece of string of the same length as the diagonal and laid it out as in the OP, I think a direct comparison could be made. If you were to look at the relationship between the area of the rectangle making the cylinder, you would see the relationship between the angle, area and length of coil, which would easily explain the observed increase in length as the angle increases. I am guessing that the piece of string would form a circle with a radius half way between the OP's spiral start and end points?

10. Yes, that does seem like a very effective way to discover the effect the angle has with a solenoid coil. It would also easily provide the length for a given number of turns.

But no thoughts on the flat, two dimensional spiral in my initial post.

11. Originally Posted by SteveC

How do I find the length of this spiral for ONE TURN ONLY The circle in the center is 7 inches in diameter, and the spiral moves outward at a constant rate. So how do I find its length?

I know of several websites that allow one to plug in a spiral's dimensions and get its length, but could you could explain the concept as well. I want to understand the theory behind it.

Thankyou
You have provided no definition of the particular spiral that you have in mind. No construction and no mathematical functcion are provided. I cannot begin to read that figure, even with magnification.

There is no general formula for the arc length of an arbitrary spiral. You can come up with a spiral to have any arc length whatever.

Without a mathematical description the best that I can suggest is to buy one of those devices used to measur distances along a curve (a road) on a map and simply run it along your curve.

There is no applicable theory for this problem since there is no description of the spiral. If you had a mathematical description of a curve say g such that g(x) were trace out the curve as a vector in the plane as x went from 0 to 1 then you would simply evaluate the integral of the norm of the derivative of g from 0 to 1.

12. Originally Posted by DrRocket
Originally Posted by SteveC

How do I find the length of this spiral for ONE TURN ONLY The circle in the center is 7 inches in diameter, and the spiral moves outward at a constant rate. So how do I find its length?

I know of several websites that allow one to plug in a spiral's dimensions and get its length, but could you could explain the concept as well. I want to understand the theory behind it.

Thankyou
You have provided no definition of the particular spiral that you have in mind. No construction and no mathematical functcion are provided. I cannot begin to read that figure, even with magnification.

There is no general formula for the arc length of an arbitrary spiral. You can come up with a spiral to have any arc length whatever.

Without a mathematical description the best that I can suggest is to buy one of those devices used to measur distances along a curve (a road) on a map and simply run it along your curve.

There is no applicable theory for this problem since there is no description of the spiral. If you had a mathematical description of a curve say g such that g(x) were trace out the curve as a vector in the plane as x went from 0 to 1 then you would simply evaluate the integral of the norm of the derivative of g from 0 to 1.
Would the function for this not simply be:

x<sup>2</sup> + y<sup>2</sup> = 7 + (0.25/3π)*θ

Where θ is the total angle subtended by the spiral at any given end-point?

13. Originally Posted by sunshinewarrior
Originally Posted by DrRocket
Originally Posted by SteveC

How do I find the length of this spiral for ONE TURN ONLY The circle in the center is 7 inches in diameter, and the spiral moves outward at a constant rate. So how do I find its length?

I know of several websites that allow one to plug in a spiral's dimensions and get its length, but could you could explain the concept as well. I want to understand the theory behind it.

Thankyou
You have provided no definition of the particular spiral that you have in mind. No construction and no mathematical functcion are provided. I cannot begin to read that figure, even with magnification.

There is no general formula for the arc length of an arbitrary spiral. You can come up with a spiral to have any arc length whatever.

Without a mathematical description the best that I can suggest is to buy one of those devices used to measur distances along a curve (a road) on a map and simply run it along your curve.

There is no applicable theory for this problem since there is no description of the spiral. If you had a mathematical description of a curve say g such that g(x) were trace out the curve as a vector in the plane as x went from 0 to 1 then you would simply evaluate the integral of the norm of the derivative of g from 0 to 1.
Would the function for this not simply be:

x<sup>2</sup> + y<sup>2</sup> = 7 + (0.25/3π)*θ

Where θ is the total angle subtended by the spiral at any given end-point?
You must have seen some information that I missed. All that I see is a spiral, and there are a lot of different spirals. I do not understand how you arrived at the coefficient for θ for instance.

14. Dr. Rocket, I did explain the dimensions of the spiral in my second post, sorry the picture ended up too small.

In the drawing I posted there is a perfect circle 7inches in diameter. The spiral completes one turn around this perfect circle. When the spiral does 1/4 of a turn, it is 1/16 of an inch away from the perfect circle, when it is halfway around it is 1/8 of an inch away from the circle, 3/4 of a turn it is 3/16 of an inch away, and the instant it completes the full turn, it is 1/4 of an inch away. It extends away from the perfect circle at this rate.

15. Originally Posted by SteveC
Dr. Rocket, I did explain the dimensions of the spiral in my second post, sorry the picture ended up too small.

In the drawing I posted there is a perfect circle 7inches in diameter. The spiral completes one turn around this perfect circle. When the spiral does 1/4 of a turn, it is 1/16 of an inch away from the perfect circle, when it is halfway around it is 1/8 of an inch away from the circle, 3/4 of a turn it is 3/16 of an inch away, and the instant it completes the full turn, it is 1/4 of an inch away. It extends away from the perfect circle at this rate.
Thanks, in that case an equation describing the spiral is

Or in cylindrical coordinates

and back to rectangular coordinates in a vector form,

You can, if you like, refer to my earlier post and from this write down an integral for the arc length that you seek.

16. Originally Posted by DrRocket
Originally Posted by SteveC
Dr. Rocket, I did explain the dimensions of the spiral in my second post, sorry the picture ended up too small.

In the drawing I posted there is a perfect circle 7inches in diameter. The spiral completes one turn around this perfect circle. When the spiral does 1/4 of a turn, it is 1/16 of an inch away from the perfect circle, when it is halfway around it is 1/8 of an inch away from the circle, 3/4 of a turn it is 3/16 of an inch away, and the instant it completes the full turn, it is 1/4 of an inch away. It extends away from the perfect circle at this rate.
Thanks, in that case an equation describing the spiral is

Or in cylindrical coordinates

and back to rectangular coordinates in a vector form,

You can, if you like, refer to my earlier post and from this write down an integral for the arc length that you seek.
Thanks for that - both correcting the mistake in my 'equation' and for extending it. I would be interested in seeing whether or not there is a general form for the function (to calculate the length of the curve on this spiral).

17. I've got a gut feeling that you can make a very close approximation by unrolling the spiral to get a triangle with base 2*pi*R and height:1/4th of an inch
R would be 3.5+1/8th: the average of the inner and the outer radius of the spiral. You are cropping the end of the curve and extending the beginning of the curve, but since the circumference of a circle depends linearly on the radius, these effects might lift each other.

18. Originally Posted by sunshinewarrior
Originally Posted by DrRocket
Originally Posted by SteveC
Dr. Rocket, I did explain the dimensions of the spiral in my second post, sorry the picture ended up too small.

In the drawing I posted there is a perfect circle 7inches in diameter. The spiral completes one turn around this perfect circle. When the spiral does 1/4 of a turn, it is 1/16 of an inch away from the perfect circle, when it is halfway around it is 1/8 of an inch away from the circle, 3/4 of a turn it is 3/16 of an inch away, and the instant it completes the full turn, it is 1/4 of an inch away. It extends away from the perfect circle at this rate.
Thanks, in that case an equation describing the spiral is

Or in cylindrical coordinates

and back to rectangular coordinates in a vector form,

You can, if you like, refer to my earlier post and from this write down an integral for the arc length that you seek.
Thanks for that - both correcting the mistake in my 'equation' and for extending it. I would be interested in seeing whether or not there is a general form for the function (to calculate the length of the curve on this spiral).
With the equation for x and y in terms of theta you can paramaterize the spiral as a curve in the plane. Then you can take the norm of the derivative of that vector function (the parameterization) and write the arc length as an integral. I have not chased that integral, but it will involve the square root of a quadratic expression in theta and may or may not be integrable in closed form. At first blush the integral looks ugly, might be elliptic.

19. Originally Posted by DrRocket
With the equation for x and y in terms of theta you can paramaterize the spiral as a curve in the plane. Then you can take the norm of the derivative of that vector function (the parameterization) and write the arc length as an integral. I have not chased that integral, but it will involve the square root of a quadratic expression in theta and may or may not be integrable in closed form. At first blush the integral looks ugly, might be elliptic.
I think with a few substitutions you can convert it to the first integral on http://en.wikipedia.org/wiki/List_of...onal_functions

Result is still quite ugly, but it's possible

20. Originally Posted by Bender
I've got a gut feeling that you can make a very close approximation by unrolling the spiral to get a triangle with base 2*pi*R and height:1/4th of an inch
Very interesting idea - the sort of thing I think forms the basis of most great maths - at least the way of thinking does.

When I thought about it for a while I was puzzled as to what, if anything, could be the flaw in the reasoning and the one that struck me was this:

In addition to the 'height' increase, the spiral also undergoes a length increase (that is not available to the triangle) by a 2*pi*r factor as it 'goes around' the original circle.

I have no idea how, if at all, you could incorporate that, but it seems to me clear that the spiral length would actually be greater than the length of the hypotenuse on your proposed triangle.

What do you think? (Still hoping for a solution in Euclidean terms, rather than one requiring the calculus...)

21. Originally Posted by sunshinewarrior
Originally Posted by Bender
I've got a gut feeling that you can make a very close approximation by unrolling the spiral to get a triangle with base 2*pi*R and height:1/4th of an inch
Very interesting idea - the sort of thing I think forms the basis of most great maths - at least the way of thinking does.

When I thought about it for a while I was puzzled as to what, if anything, could be the flaw in the reasoning and the one that struck me was this:

In addition to the 'height' increase, the spiral also undergoes a length increase (that is not available to the triangle) by a 2*pi*r factor as it 'goes around' the original circle.

I have no idea how, if at all, you could incorporate that, but it seems to me clear that the spiral length would actually be greater than the length of the hypotenuse on your proposed triangle.

What do you think? (Still hoping for a solution in Euclidean terms, rather than one requiring the calculus...)
This is reflected in the base of the triangle: if you take the inner circumference, you get an underestimation, if you get the outer circumference, you get an overestimation. In this particular example, both are already reasonably close together. Taking the linear average is on first sight not unreasonable, since the circumference is linearly dependant on the radius, and the radius changes linearly. I don't have to time to check how big the error (if any) would be at the moment

22. Originally Posted by Bender
Originally Posted by sunshinewarrior
Originally Posted by Bender
I've got a gut feeling that you can make a very close approximation by unrolling the spiral to get a triangle with base 2*pi*R and height:1/4th of an inch
Very interesting idea - the sort of thing I think forms the basis of most great maths - at least the way of thinking does.

When I thought about it for a while I was puzzled as to what, if anything, could be the flaw in the reasoning and the one that struck me was this:

In addition to the 'height' increase, the spiral also undergoes a length increase (that is not available to the triangle) by a 2*pi*r factor as it 'goes around' the original circle.

I have no idea how, if at all, you could incorporate that, but it seems to me clear that the spiral length would actually be greater than the length of the hypotenuse on your proposed triangle.

What do you think? (Still hoping for a solution in Euclidean terms, rather than one requiring the calculus...)
This is reflected in the base of the triangle: if you take the inner circumference, you get an underestimation, if you get the outer circumference, you get an overestimation. In this particular example, both are already reasonably close together. Taking the linear average is on first sight not unreasonable, since the circumference is linearly dependant on the radius, and the radius changes linearly. I don't have to time to check how big the error (if any) would be at the moment
So if I understand you correctly you are talking about two circumferences? One with 7" radius and one with 7.25" radius?

23. Would the outer spiral not have an average radius of th of an inch + that of the inner circle? If so, would the circumference of this circle not give the length of the outer spiral?

24. Originally Posted by sunshinewarrior
Originally Posted by Bender
Originally Posted by sunshinewarrior
Originally Posted by Bender
I've got a gut feeling that you can make a very close approximation by unrolling the spiral to get a triangle with base 2*pi*R and height:1/4th of an inch
Very interesting idea - the sort of thing I think forms the basis of most great maths - at least the way of thinking does.

When I thought about it for a while I was puzzled as to what, if anything, could be the flaw in the reasoning and the one that struck me was this:

In addition to the 'height' increase, the spiral also undergoes a length increase (that is not available to the triangle) by a 2*pi*r factor as it 'goes around' the original circle.

I have no idea how, if at all, you could incorporate that, but it seems to me clear that the spiral length would actually be greater than the length of the hypotenuse on your proposed triangle.

What do you think? (Still hoping for a solution in Euclidean terms, rather than one requiring the calculus...)
This is reflected in the base of the triangle: if you take the inner circumference, you get an underestimation, if you get the outer circumference, you get an overestimation. In this particular example, both are already reasonably close together. Taking the linear average is on first sight not unreasonable, since the circumference is linearly dependant on the radius, and the radius changes linearly. I don't have to time to check how big the error (if any) would be at the moment
So if I understand you correctly you are talking about two circumferences? One with 7" radius and one with 7.25" radius?
I thought it was 3.5" and 3.75" Radius, but that's a detail.

The line integral does indeed result in the integral I pointed out, but the triangle approximation I suggested comes very close with a relative error of .
It could be a numeric error, but the equations don't look that poorly conditioned and with an outer radius of 20", the error increases to , so the solution is probably not exact.

25. Originally Posted by Bender
Originally Posted by DrRocket
With the equation for x and y in terms of theta you can paramaterize the spiral as a curve in the plane. Then you can take the norm of the derivative of that vector function (the parameterization) and write the arc length as an integral. I have not chased that integral, but it will involve the square root of a quadratic expression in theta and may or may not be integrable in closed form. At first blush the integral looks ugly, might be elliptic.
I think with a few substitutions you can convert it to the first integral on http://en.wikipedia.org/wiki/List_of...onal_functions

Result is still quite ugly, but it's possible
Yep you can do the intnegral as you suggest. And yep it is as ugly as I thought it would be -- truly hideous. I am not going to bother to try to put this mess into Tex, since it is not very illuminating, although it is an exact closed form expression for the arc length, involving only logs, square roots, and rational functions in pi.

Just looking at that godawful expression, I am convinced that sunshinewarriors hope for an exact solution using only basic algebra is doomed.

26. Originally Posted by DrRocket

Just looking at that godawful expression, I am convinced that sunshinewarriors hope for an exact solution using only basic algebra is doomed.
We're doomed? Don't panic!

(Sorry - old Brit joke there).

Any other true mathematicians (as opposed to fake/dilletantes like me) here who might have the time to take it on - just to wow us, like?

27. Originally Posted by SteveC
Believe this or not, a spring does not really change in length much until you get past 45 degrees vertically.

In other words if you have a round flat coil one rotation, and you keep it under 45 degrees vertically, the length will remain about the same as you open the spring up to about 45 degrees. After that it will tighten the circle.
I just wrapped 4 turns around a rubber band at maybe a 10 degree tilt, then I upped it to 45 degrees and it was very close to the same. That is somewhat strange. Once it gets past 45 degrees, it starts making extreme changes to the number of turns. It would be interesting to see a graph of the angle of inclination versus the loss of rotations. I bet the graph would not be linear at all, because from 0 to 10 degrees we get basically no change, and from 80 to 90 degrees we lose all the rotations completely.

I had done these experiments to show and explain how I learned about electrons, traveling through matter, and how they would move around atoms in matter that were in their way.

They don't let me post this on the physics forum because they say it is not in line with modern science. Which I am actually happy about.

That is why I mention electrons in the movie. Because I was demonstrating that an electron with a similar force moving it, while moving through matter. Would move more quickly along a larger spiral path then the tighter spiral path.

This is what I was taught, was the basis for electrons moving in larger spiral paths causing a higher frequency to be generated in matter. Because they were passing a larger number of atoms per second. Because they experience less resistance to movement in the larger loop.

The theory I was taught was that light in the blue spectrum took a larger spiral path, that caused a straighter angle in its path, then red light.

That allows blue light to move deeper then green or red light.
X-rays take an even larger straighter spiral path and can go much deeper into material. Since something traveling along the larger diameter spiral, moves a greater distance more quickly as demonstrated by the spiral experiment, you can see how it would create a higher frequency in the material.

I was taught that it was this reason that the eye was able to tell the difference in color. Because red light only hit the surface of the protrusions in the eye. Green and blue were able to go much more deeply.

All the spirals paths are the same length in the movie.

http://www.Rockwelder.com/Flash/Colo...oroflight2.htm

http://www.rockwelder.com/Flash/Colo...orofLight1.htm

Sincerely,

William McCormick

28. Originally Posted by KALSTER

If you were to draw a rectangle, connect the diagonal corners and rolled it into a cylinder, the diagonal would form a spiral. If you then took a piece of string of the same length as the diagonal and laid it out as in the OP, I think a direct comparison could be made. If you were to look at the relationship between the area of the rectangle making the cylinder, you would see the relationship between the angle, area and length of coil, which would easily explain the observed increase in length as the angle increases. I am guessing that the piece of string would form a circle with a radius half way between the OP's spiral start and end points?
That is how they make spiral duct from flat stock. That is cool.

But when you are going up and around a spiral, it is not a straight line in either direction anymore. There is a double arc visible in reality.

That straight line on the paper, is bent when you roll it. When you do that you change things.

It is like using a break or brake bending device on it, thousands and thousands of times. But instead of bending it perpendicular to its length and making a circle, you are bending it at what ever degree matches your rise. And you get a helix. Or a thread pattern.

I make threads on a lathe all the time. And I know that they are not straight lines at all. They cannot be, because that is impossible. They do though however match the rise and run of the diagonal drawn on the paper, which shows the shortest path around a spiral. Very refreshing thinking.

If you were to wrap the paper around a thread it would match the thread pattern. I actually turned a piece of paper on the lathe with a pen instead of a cutter. And it does certainly create very straight lines when you unroll the paper.

When you machine threads and if you think about the different threads possible either course or fine. The lathe will cut a single thread per revolution of the lathe. Either course or fine. It is hard to measure the travel of the bed into the calculation unless you use the diagonal paper trick. The difference would be very small.

When I make spiral stair cases they are at least 60 inches in diameter, that means that the circumference is about 188 inches. So if I am going from the top of the lowest baluster to the highest baluster that is about 96 inches in most houses.

So when I create the spiral although the spiral staircase is on about, a 45 degree pitch. If drawn on flat paper, it would create a rectangle 188 inches by 96 inches. So I will lose in length only about 23 inches. When pulling it. We normally leave them about an extra four feet long and roll them into a circle. And then we just stretch them.

But it is hard to believe that you can stretch the spring 96 inches and only lose 23 inches.

When we use flat stock or normal iron handrail. We roll it into a circle the diameter of the diagonal of the staircase diameter. From the top of the lowest baluster to the height of the baluster halfway around the staircase from the lowest baluster.
When you pull up on the handrail as you would to open or stretch a spring, the diameter reduces almost magically. I understand it is happening. But wow, it is wild to explain.

Sincerely,

William McCormick

29. Mod note: William I remind you this is a mathematics forum. If you want to discuss engineering, please find another sub-forum. Preferably on another site

30. Guitarist: I would recommend that you take the strongest action possible against William McCormick. He is a far too dangerous to be let loose in any scientific community – or indeed anywhere on earth. The sooner he departs, the better for all of humanity. I would dearly love to have a countdown clock of his remaining lifespan so I could watch it every day and follow the countdown with relish.

31. Originally Posted by JaneBennet
Guitarist: I would recommend that you take the strongest action possible against William McCormick. He is a far too dangerous to be let loose in any scientific community – or indeed anywhere on earth. The sooner he departs, the better for all of humanity. I would dearly love to have a countdown clock of his remaining lifespan so I could watch it every day and follow the countdown with relish.
I agree with you in principle. But don't sugar-coat it so much. :-)

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