# Spiral equation

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

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

• How is the yellow curve related to gray arcs on pictures? It doesn't seem to pass through endpoints of those arcs... Commented Jul 22, 2020 at 15:10
• 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.
– alex
Commented Jul 22, 2020 at 15:15

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.
• 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)