Limit definition of curvature and torsion

Example given is for curvature in the osculation plane rotation $\theta$ :

Arc $ s = 2 R \theta $ in $ \mathbb R^2$

Direct Euclidean distance $ d = 2 R \sin \theta$

Series approximation for third order

$$ s/R = 2 \theta;\, d/R= 2 \sin \theta \approx 2(\theta - \theta^3/3!)= 2(s/2R- (s/R)^3/8\cdot3!) $$

$$ d \approx s - s^3/(24 R^2 )$$

$$ \frac1R = \kappa \approx \sqrt \frac{ 24 (s-d)}{s^3}$$

the same as your result. You see that we dealt with infinitesmal lengths ( that later on tend to zero) as small finite lengths that can be sketched to visible proportions. [Incidentally I read somewhere it was similarly handled by Leibnitz during earliest stages of calculus].

We used $ t' = \kappa\, n $ in Frenet-Serret frame

Similarly we can now develop

$$ b'= \tau \,n $$

in the plane of normals (principal and bi- normal)

Let the instantaneous radius of torsion be $\rho$, we have torsion

$$ \tau = \frac{d \theta }{ds} = \frac{\sin \psi }{\rho} $$

Length $ dl $ extends by twisting to $ ds$ in normal plane so that $ \cos \psi = dl/ds$

$$ \sin \psi \approx 1 - ( dl /ds)^2/ 2 ;\,\, \tau = \frac{1 - ( dl /ds)^2/ 2 }{\rho} $$

You can write/define using your nomenclature

$$ ds = s(P,Q), dl = d(P,Q). $$