Polar equation for a unique growth spiral

[Sluggonomics]

Hi,
I am trying to derive a polar equation which defines the curvature of a spiral whose radius grows proportionally with its arc length.
Ex: for any given 1cm arc length, the radius grows at a factor of 1.1

I am having a tough time accomplishing this and I am very eager to see the solution.
I would like to define radius as a function of theta, related using these variables:

k : growth factor (ex: 1.1 cm^-1) k>1
a : initial radius
Theta : angle from initial radius

Seemed so simple but not so much...
Thank you!

 

[raquelc]

If you look at it as a 'line' on the form y=mx+c, the gradient is the growth factor, the angle will be your theta and the initial radius your y-intercept.

 

[Sluggonomics]

Hmm I could be wrong but that sounds like an archimedean spiral R=m*(theta)+a, which does not grow proportionally with arc length

 

[Dr.Peterson]

It isn't quite clear to me exactly what your description means. One major issue is that you've said nothing about how the angle theta changes along the curve, so as far as I can tell, the curve could be a straight radial line at fixed theta. Another is that "any given 1 cm arc length" is not the same as "its [total?] arc length".

It may help if you sketch what you think it might look like and show on the sketch how you would judge whether it fits your description. You might also explain the real situation (if any) that motivates this goal.

 

[Sluggonomics]

Excellent point, a sketch is in order!


Apologies for a botched first attempt on the same page, but this example spiral is starting with an initial radius of 1.25 inches (polar grid radius is divided into 1/4" units) and then I used a compass to draw the chord (approximating arc length) by seeing where the 1" compass line intersected the radius grid for a next radius value of (1.25)*1.1^n (nth chord). I described this as growing proportionally but I think I should have said exponentially, my mistake.

If I'm not mistaken, the ultimate equation would be derived from the limit of an approximation like this as the chords become infinitesimally small.

As for the real situation, imagine a shaped pulley being tugged on by a string and sprung to oppose the tug. I would like the pulley's shape to have a varying radius but be self-similar in the sense that for each small unit of string that is pulled, the radius increases by the same factor (1.1 over 1" in my example sketch)

 

[raquelc]

Excellent point, a sketch is in order!
View attachment 19397

Apologies for a botched first attempt on the same page, but this example spiral is starting with an initial radius of 1.25 inches (polar grid radius is divided into 1/4" units) and then I used a compass to draw the chord (approximating arc length) by seeing where the 1" compass line intersected the radius grid for a next radius value of (1.25)*1.1^n (nth chord). I described this as growing proportionally but I think I should have said exponentially, my mistake.

If I'm not mistaken, the ultimate equation would be derived from the limit of an approximation like this as the chords become infinitesimally small.

As for the real situation, imagine a shaped pulley being tugged on by a string and sprung to oppose the tug. I would like the pulley's shape to have a varying radius but be self-similar in the sense that for each small unit of string that is pulled, the radius increases by the same factor (1.1 over 1" in my example sketch)

My approach of r=1.1*theta+a was for proportional growth. Could you try r=a*theta^1.1?that will start at (0,0), though

 

[Dr.Peterson]

Arc length can be tricky to work with; I'm pretty sure this will require a differential equation (and might not be solvable). I haven't tried yet.

How much calculus do you know?

 

[Sluggonomics]

My calculus skills have a decade of rust and were never shiny in the first place, but I understand the concept and can manage my way through basic stuff.

I know it will involve general arc length formula

But this is where I start to fizzle out from limited skillset.

By considering the spiral I think it should be solvable for a bounded range of theta, eventually reaching a radial asymptote

 

[Cubist]

I think the real life problem (see the bottom of post #5) is different to the problem asked in post#1. I will explain this below:-

I started thinking about how I would solve this numerically. I considered splitting the curve into triangles like this:-



The current string contact point is at C. The string has already been pulled by large distance L, and the dark triangle represents pulling the string an extra small distance "l" so that D becomes the next string contact point and the radius changes from r1 to r2 (both r lengths are given by your equation). Using the cosine rule angle "a" can then be calculated. This angle is added to a running total to generate a table of angle vs radius.

BUT, for this diagram, angle b must be under 90° otherwise the horizontal string would not initially make contact at point C. If D is higher than point C then the string would initially be at the height of D. This would mean the radius will be greater than desired at this string length. And I believe that angle b will be greater then 90° for your particular equation. Actually this can be seen by looking at your sketch in post#5 (consider the tangents to your sketch).

So this opens a slight can of worms. A numeric solution would need to skew the triangle slices to the left, as in the following diagram...



Initially C is the contact point (and D must be the same height or lower) and after the string is pulled by a small length then D becomes the next contact point and the vertical distance OE would become r2. (Any part of the curve to the sides of OCD must also be lower than C and D).

Before going any further, would a numeric solution be suitable for your needs anyway? Is it even OK to assume the string will be horizontal (is it is fixed to something quite far away)?

 

[Sluggonomics]

I think the real life problem (see the bottom of post #5) is different to the problem asked in post#1. I will explain this below:-

I started thinking about how I would solve this numerically. I considered splitting the curve into triangles like this:-

View attachment 19451

The current string contact point is at C. The string has already been pulled by large distance L, and the dark triangle represents pulling the string an extra small distance "l" so that D becomes the next string contact point and the radius changes from r1 to r2 (both r lengths are given by your equation). Using the cosine rule angle "a" can then be calculated. This angle is added to a running total to generate a table of angle vs radius.

BUT, for this diagram, angle b must be under 90° otherwise the horizontal string would not initially make contact at point C. If D is higher than point C then the string would initially be at the height of D. This would mean the radius will be greater than desired at this string length. And I believe that angle b will be greater then 90° for your particular equation. Actually this can be seen by looking at your sketch in post#5 (consider the tangents to your sketch).

So this opens a slight can of worms. A numeric solution would need to skew the triangle slices to the left, as in the following diagram...

View attachment 19453

Initially C is the contact point (and D must be the same height or lower) and after the string is pulled by a small length then D becomes the next contact point and the vertical distance OE would become r2. (Any part of the curve to the sides of OCD must also be lower than C and D).

Before going any further, would a numeric solution be suitable for your needs anyway? Is it even OK to assume the string will be horizontal (is it is fixed to something quite far away)?

Fantastic reply, and you are correct on all counts:

It is okay to assume the string is horizontal as it will be fixed far enough away that any angular change will be negligible.

The string would indeed contact the pulley skewed to the left, as determined by the tangent. I have been working on this project in steps of gradually increasing complexity and hadn't wanted to bite off more than I could chew, hoping to compensate for that after finding the "simpler" formula. Seeing as it's not so simple anyways, maybe best to aim for the correct shape right from the get-go! A numerical solution that addresses the extra complication is most welcome, but I must admit the nerd deep down within me still hopes for a general formula.

For either the "simpler" case or the "true" shape, or both

Big thank you to everyone who's taking the time to consider this, I have found it most stimulating and am very excited to see the solutions.