Spirals...

[Dr. Zimski]

How would one measure the length of a spiral? I tried figuring it out with nothing but a piece of paper and a pencil, but could figure nothing. I have a feeling it might have something to do with log, cos, sin, and/or tan.

 

Thank's in advance for your help, and I'm sorry if I sound stupid.

[ecoli]

You could do it by integrating the arc length. You would have to measure the polar coordinates of the spiral, though.

 

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

[herpguy]

How would one measure the length of a spiral? I tried figuring it out with nothing but a piece of paper and a pencil' date=' but could figure nothing. I have a feeling it might have something to do with log, cos, sin, and/or tan.

 

Thank's in advance for your help, and I'm sorry if I sound stupid.[/quote']

 

Take a string, put it on the spiral, and measure the string that was placed on the spiral. I'm also sorry if I sound stupid.

[Sisyphus]

Spirals are tricky because they're composed of two completely independant motions: the radial and rotational, each of which can vary freely of the other. (To trace a spiral, spin a ruler around one point, and move your pencil along the length of the ruler.) As a result, there are many different kinds of spirals, and I don't think there's one equation which can represent its length. However, I think you can generally have a good approximation if you divide it into segments of equal angular distance, then use the average radius over each segment as if it were the radius of the arc of a circle of equal angular distance, thereby approximating the spiral with a series of segments of circles of different radii.

[ecoli]

Spirals are tricky because they're composed of two completely independant motions: the radial and rotational, each of which can vary freely of the other. (To trace a spiral, spin a ruler around one point, and move your pencil along the length of the ruler.) As a result, there are many different kinds of spirals, and I don't think there's one equation which can represent its length. However, I think you can generally have a good approximation if you divide it into segments of equal angular distance, then use the average radius over each segment as if it were the radius of the arc of a circle of equal angular distance, thereby approximating the spiral with a series of segments of circles of different radii.

 

You can approximate arc length rather accurately, I guess the worst part is that you'd need a lot of sample points to get a good measure.

[yammers]

I'm no math guy, but I think this looks logical...

 

Well it might be wrong, but it was fun trying.

 

Please forgive me if otherwise, but I'll assume that your spiral was a perfect coil with the radius beginning at 0 degrees and spiralling down through-out 3 spirals and ending at 360 degrees. Now, when you look straight down on this, it will appear as a circle O (Let's say 2 ft dia.) . When you view it from the side it should look like a zig zag (not the paper kind) and let's say 3' tall; connect the dots.

 

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So, I figure you'd used Pythagorus Theorum (AA + BB = CC), but you'd use the circ instead of the horizontal length x 2 of a complete 360.

 

 

(Total height / 3 spirals) squared + (dia x pie) squared = 360 length squared

 

3/3 sq + 6.28 sq = lgth sq

 

1 + 39.44 = 40.44

 

6.36' per 360 degrees

 

Total length of a spiral 2'wide x 3' tall with 3 complete 360's should be = 19.08'

 

I don't know...maybe...

[yammers]

It looks as though I should have confirmed the definition of spiral first...

Oh well, it won't be the last time I make a mistake.

[Dr. Zimski]

It looks as though I should have confirmed the definition of spiral first...

Oh well' date=' it won't be the last time I make a mistake.[/quote']

Yeah, that, what you were talking about was a coil. To figure out a coil is just like a triangle: hypotenuse^2 = (h^2) + [pi(d)]^2, where the hyp. is the length of the coil, h is the hieght of the coil, and [pi(d)] is the circumference.

[EvoN1020v]

I suggest you to get a circle graphic paper and put dots where x=y, then you are creating a spiral. If you use Pythagorus Theorem, you are assuming that the lines are straight from a point to point. That's not true, because a spiral have NO straight lines. So you have to to use [math]2r\pi[/math] (circumstance), but again comes a problem: The spiral have NO circles. Hence, an idea just popped in my head: Use semi-circle. It's hard to explain, but, on a spiral you can divide a spiral into semi-circle segments and measure the curve side of the semi-circle, and add all the measurements for the whole length of the sprial.

 

Hope this helps.

[Dr. Zimski]

I suggest you to get a circle graphic paper and put dots where x=y' date=' then you are creating a spiral. If you use Pythagorus Theorem, you are assuming that the lines are straight from a point to point. That's not true, because a spiral have NO straight lines. So you have to to use [math']2r\pi[/math] (circumstance), but again comes a problem: The spiral have NO circles. Hence, an idea just popped in my head: Use semi-circle. It's hard to explain, but, on a spiral you can divide a spiral into semi-circle segments and measure the curve side of the semi-circle, and add all the measurements for the whole length of the sprial.

 

Hope this helps.

I know that a spiral has no strait lines, what I was talking about in my other post was a coil, like a metal spring. A spiral is completely different.

[woelen]

Why all these difficult things?

 

If you have a mathematical description of your spiral (x, y, z) as function of running parameter t, then integrate along the path the length of the curve. This may require numerical methods of integration, but fortunately, integration is very easy and stable numerically.

 

If you do not have a mathematical description, then determine many points (x, y) (or (x, y, z) is it is 3D) and estimate the length of the spiral, simply by taking the length of all small line segments. Better approximations are obtained by interpolating quadratically or cubically between the data points and integrating that.

[yammers]

Yeah, that, what you were talking about was a coil. To figure out a coil is just like a triangle: hypotenuse^2 = (h^2) + [pi(d)]^2, where the hyp. is the length of the coil, h is the hieght of the coil, and [pi(d)'] is the circumference.

 

Couldn’t one use your method, but instead of simply measuring the diameter of the circle, take the average of the spirals? Assuming that the spiral starts at 0 degrees, and ends at 360 degrees; like calculating the average diameter of a cone, only with the spiral you’d factor in the number of spirals instead of just each end. By factoring in the number of revolutions, the curve should be accounted for. No?

As I stated before, I’m not a math guy…Just a hack, hacking away, so go easy on me…

[Dr. Zimski]

Couldn’t one use your method' date=' but instead of simply measuring the diameter of the circle, take the average of the spirals? Assuming that the spiral starts at 0 degrees, and ends at 360 degrees; like calculating the average diameter of a cone, only with the spiral you’d factor in the number of spirals instead of just each end. By factoring in the number of revolutions, the curve should be accounted for. No?

As I stated before, I’m not a math guy…Just a hack, hacking away, so go easy on me… [/quote']

That method wasn't for spirals, but for coils (like a metal spring), assuming that the coil is (for lack of better word) "rising" at a perfectly even rate with even intervals. If you were to draw a curve on a paper tube that "rose" at an even rate, it should, if you unwrap it, end up looking like a right triangle if you unwrap one section. There, you can just use the pathagorum(sp?) theorum.

 

But a spiral, on the other hand is totaly different. It's hard to explain, but I figure the best way to desribe it is a 2d curve that goes on out forever but never touches itself.

[yammers]

If you were to draw a curve on a paper tube that "rose" at an even rate, it should, if you unwrap it, end up looking like a right triangle if you unwrap one section.

 

OK, now, if you were to draw a curve on a paper funnel (like the ones at gas stations) that "rose" ever increasing (spiral), it should, if you unwrap it, end up looking like a right triangle if you unwrap one section.

 

I have been going by this definition, perhaps it's wrong for the purpose of this discussion:

Spiral- 1. a plane curve formed by a point moving around a fixed central point in a continuosly increasing or decreasing arc. Gage Canadian Dictionary

 

I am more and more convinced that this would work out...But maybe I'm missing something.

[Dr. Zimski]

OK' date=' now, if you were to draw a curve on a paper funnel (like the ones at gas stations) that "rose" ever increasing (spiral), it should, if you unwrap it, end up looking like a right triangle if you unwrap one section.

 

I have been going by this definition, perhaps it's wrong for the purpose of this discussion:

[b']Spiral[/b]- 1. a plane curve formed by a point moving around a fixed central point in a continuosly increasing or decreasing arc. Gage Canadian Dictionary

 

I am more and more convinced that this would work out...But maybe I'm missing something.

I meant cylinder, not cone.

 

Yes, that is a spiral. But what I mentioned before (not in my main question though), was completely different from a spiral.

 

Never mind anyways, I don't need to know how to figure the length of a spiral anymore.

[cosine]

Gr I still want to know. I might figure it out by hand soon enough. It wouldn't be too hard. Make an infinite sum of the slices of an arc using the formula arc = theta * radius, and then just compare it to how a reimann sum is expressed as an integral, and do the same with the your infinite sum for arclength. Thats similar to how I derived the arclength integral for a function in cartestian co-ordinates a while back, except then I used the pythagorean theorem to measure the small little lengths.

[yammers]

I meant cylinder' date=' not cone.

 

Yes, that is a spiral. But what I mentioned before (not in my main question though), was completely different from a spiral.

 

Never mind anyways, I don't need to know how to figure the length of a spiral anymore.[/quote']

 

Dr. Zimski, I assume that you either know the answer to your original question, or you don't care anymore...In what ever case, you did ask an interesting question that is still waiting to be solved by a credible source, and verified. I only offered suggestions in hopes of stimulating the discussion further, and because I thought I might know, but I too am waiting for verification.

 

Does anyone else know? Cosine have you worked it out?

[cosine]

Dr. Zimski' date=' I assume that you either know the answer to your original question, or you don't care anymore...In what ever case, you did ask an interesting question that is still waiting to be solved by a credible source, and verified. I only offered suggestions in hopes of stimulating the discussion further, and because I thought I might know, but I too am waiting for verification.

 

Does anyone else know? Cosine have you worked it out?[/quote']

Lol, now I must finally do it. I'll work it out now... (post in a few)

 

Hmm okay i just worked it out on paper, i'm uploading it now. (thanks digimemo!)

[cosine]

Darn, how do I upload an image?

 

Edit, nevermind, here it is:

 

(Excuse the writing lines in the background, they shift from my digipad to the computer, though my writing skill isn't that great anyway. And on the final line Theta_0 became Theta_2 because its 4 in the morning..., and its the digipad's fault that some L's and one's look like r's)

[Dr. Zimski]

Dr. Zimski' date=' I assume that you either know the answer to your original question, or you don't care anymore...In what ever case, you did ask an interesting question that is still waiting to be solved by a credible source, and verified. I only offered suggestions in hopes of stimulating the discussion further, and because I thought I might know, but I too am waiting for verification.

 

Does anyone else know? Cosine have you worked it out?[/quote']

Meh, I don't know, I'm just not really interested anymore.

 

[EDIT:] ^^^Well, there you have it. Good job.

[cosine]

Meh' date=' I don't know, I'm just not really interested anymore.

 

[EDIT:'] ^^^Well, there you have it. Good job.

Thankyou!