So here's a little background for the question:(adsbygoogle = window.adsbygoogle || []).push({});

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.

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# Spiral arc length question

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