# Arc length parametrisation of a logarithmic spiral

$$\gamma(t)=(ae^{bt}cos(t),ae^{bt}sin(t),0)$$

(a) Find the arc-length parametrisation.

So I think that the parametrisation is $$\gamma(s)= \bigg((\frac{bs}{\sqrt{b^2+1}})cos(\ln\bigg(\frac{bs}{a\sqrt(b^2+1)}\bigg)),(\frac{bs}{\sqrt{b^2+1}})sin(\ln\bigg(\frac{bs}{a\sqrt(b^2+1)}\bigg)),0\bigg)$$

(b) Find the curvature and torsion.

The torsion $$\tau(s)=0$$ which is clear because it's a plane curve. But the rest is getting a bit confusing.

(c) Find the Frenet reference.

I know that this is $$T(s)=\gamma'(s) , N(s)=\frac{T'(s)}{\kappa(s)}$$ and $$B(s)=T(s)\times N(s)$$ but because I am unsure of my $$\gamma(s)$$, I'm not too certain on the rest.