how to find $t$ and $r$ of Archimedean spiral with known spiral length?

Question

I have a generic archimedean spiral $r=a\cdot\frac{t}{\pi}$ and have a spiral length $s$.

by knowing the formula for the spiral, and knowing that the spiral has no rotation applied, and knowing the spiral length how can $r$ and $t$ be calculated?

Answer 1

EDIT

Derivate wrt polar angle t.

$$r= at/\pi,\quad r'= a/\pi\;$$

Arc length

$$ L= \int_0^ T \sqrt{ r^2+r^{'2}} dt =(a/\pi)\int_0^T \sqrt{1+t^2}dt$$ which is evaluated using hyperbolic functions.

$$ L= ( a/\pi)\frac12 (T\sqrt{1+T^2} +ArcSinh T )$$

This result helps to numerically find $T$ when $L$ is given (no need r anymore) with an iterator.