Solving for Spiral Arc Length: A Scientist's Approach

[alpha754293]

So here's a little background for the question:

I have an arc that covers 3/4s of a circle (so it's not quite a full circumference) such that the radius from the center of the arc varies with respect to the angle (dR/d(theta)) (and it can be either positive or negative, but not constant).

I am trying to figure out what would be the equation that would be able to calculate the angle required to cover a certain distance of the varying radius arc.

I've built an Excel spreadsheet that approximates the solution by using fixed arclength intervals of 0.1, and I've computed the change in radius and the change in angle required to cover that arc length interval.

So, in this example, here are the initial values:

R_initial = 6 inches
R_final = 2 inches
theta_inital = 0 radians
theta_final = 3*pi/2 radians

(so it has a d(R)/d(theta) rate of 0.8488 inches per radian)

And the way that I've calculate it so far is:

d(L) = d(R) * d(theta)
R_(i+1) = R_i - (d(R)/d(theta)*theta_i)
theta_(i+1)=dL/R_(i+1)

I know that as the radius decreases (as a function of theta), the d(theta) will have to increase (in order to "travel" or "cover" the same distance) for a given interval length.

So my question is what's the angle required to traverse 3 inches of arclength?

And how I can make the equation more generic so that given any R_initial and R_final, and total arc length (L), it will tell me what's the angle required to accomplish this?

R_initial=6 inches
theta_initial = d(L)/6 = 0.1/6 = 0.01666 radians

R_(i+1) = R_initial - (d(R)/d(theta) * theta_initial)
theta_(i+1) = d(L)/R_(i+1) = d(L)/(R_initial - (d(R)/d(theta)*theta_initial))
R_(i+2) = R_(i+1) - (d(R)/d(theta) * theta_(i+1))
theta_(i+2) = d(L)/R_(i+2) = d(L)/(R_(i+1) - (d(R)/d(theta) * theta_(i+1)))
etc...

I'm trying to find a better, more generic way of computing it without having to compute all of the intermediary steps, which will also given me a more exact solution rather than an approximation.

Help would be greatly appreciated. Thank you in advance!

Oh...and P.S. The way that I've been able to answer the question of how much angle do I need to accomplish a certain distance of travel is by summing up all of the individual pieces until I get the length and the doing the same summation on my angle column (in Excel) to find that out.

But I'm trying to find an equation that will do that compute it directly. Thanks.

 

[lurflurf]

If I understand you correctly the formula you need is

$$\mathrm{distance}=\int_a^b \! \sqrt{[ r (\theta) ]^2+[ r^\prime (\theta) ]^2} \, \mathrm{d}\theta %\\ \text{where} \\r(\theta)=6-\frac{8\theta}{3\pi} \\ a=0 \\ b=x<\frac{3\pi}{2}$$
You can work the equation out, but it cannot be solved in closed form, you could use an iterative method.

 

[alpha754293]

If I understand you correctly the formula you need is

$$\mathrm{distance}=\int_a^b \! \sqrt{[ r (\theta) ]^2+[ r^\prime (\theta) ]^2} \, \mathrm{d}\theta %\\ \text{where} \\r(\theta)=6-\frac{8\theta}{3\pi} \\ a=0 \\ b=x<\frac{3\pi}{2}$$

You can work the equation out, but it cannot be solved in closed form, you could use an iterative method.

Dumb question, but what's r'(theta)?

And am I really only limited to solving this via an iterative method?

P.S. I think that the "where" text didn't show up properly... (because I can only see it as TeX, but no other way). (Sorry, I'm sort of new to the whole TeX commands/method of showing math equations. Thank you once again!

 

[lurflurf]

r' is the derivative

$$\mathrm{distance}=\int_a^b \! \sqrt{[ \mathrm{r} (\theta) ]^2+\left[ \dfrac{d}{d\theta}\mathrm{r}(\theta) \right]^2} \, \mathrm{d}\theta $$

the where stuff was a note to myself, it was not supposed to show up, but I should have removed it at the end.

 

[alpha754293]

r' is the derivative

$$\mathrm{distance}=\int_a^b \! \sqrt{[ \mathrm{r} (\theta) ]^2+\left[ \dfrac{d}{d\theta}\mathrm{r}(\theta) \right]^2} \, \mathrm{d}\theta $$

the where stuff was a note to myself, it was not supposed to show up, but I should have removed it at the end.

Oh...gotcha. Thanks! ;) And it's still only solvable in the open form only, correct? Which means that it is an iterative-solution only? (Just checking - to make sure that I haven't missed anything and that what I did with the iterative solution that I've wrote about in my initial question IS actually the best solution that exists.) Thanks.

 

[lurflurf]

if I have not made an error we must solve an equation involving

$$\mathrm{f}(x)=\int \sqrt{1+x^2}\mathrm{d}x=\dfrac{1}{2}(x\sqrt{1+x^2}+\sinh^{-1}(x)) $$
which cannot be explicitly solved in terms of common functions

 

[lurflurf]

Well those would be two methods
1) approximate the integral with a function we can solve exactly
2)do the integral exactly and solve the equation approximately

I am inclined to suggest 2) is better

 

[alpha754293]

Well those would be two methods
1) approximate the integral with a function we can solve exactly
2)do the integral exactly and solve the equation approximately

I am inclined to suggest 2) is better

So, would I be correct in saying that the iterative approach that I've built using an Excel spreadsheet is one of the possible ways for me to solve it? I had troubles late last night trying to come up with the differential equation to integrate, but it looks like that it's going to be an iterative approximate solution no matter which way, whether it's 1) or 2).

Thank you!