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Considering concentric arcs, of equal developed length, whose start point is aligned:

enter image description here enter image description here

I am looking for the equation of the spiral passing through the end points.

Some help to solve this problem will be welcome!


Edit: The result

enter image description here

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  • $\begingroup$ How is the yellow curve related to gray arcs on pictures? It doesn't seem to pass through endpoints of those arcs... $\endgroup$
    – CiaPan
    Commented Jul 22, 2020 at 15:10
  • $\begingroup$ Sorry, I do not have the equation yet. The yellow spiral is a spiral of equation Xt = -5*t*cos(t) and Yt = -5*t*sin(t). I put this screenshot to show the equation format I am looking for. $\endgroup$
    – alex
    Commented Jul 22, 2020 at 15:15

1 Answer 1

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In polar coordinates, every arc starts at $\theta=0$ and ends at $\theta=L/r$, where $L$ is the length of each arc and $r$ is the radius for respective arc. So this is the equation: $$\theta=L/r.$$ In Cartesian coordinates: $$(x,y) = \left(r\cdot\cos\frac Lr,\, r\cdot\sin\frac Lr\right)$$ for $0 < r < \infty.$

The spiral is called hyperbolic spiral, or a reciproke spiral – see my post Does the spiral Theta = L/R have a name? and the answer to it.

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  • $\begingroup$ For those reading the answer, I would just like to add that the x-axis is vertical up, and y is pointing to the right. $\endgroup$
    – Andrei
    Commented Jul 22, 2020 at 21:00
  • $\begingroup$ Thank you very mutch, I solved it with t1=Rmin=0.01, t2=Rmax=0.05, xt=1000*t*sin(0.001*length/t), yt=1000*t*cos(0.001*length/t) $\endgroup$
    – alex
    Commented Jul 23, 2020 at 6:52

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