Limit definition of curvature and torsion Given two point, $P$ and $Q$, lying on a curve $\gamma: \mathbb{R} \to \mathbb{R}^3$, curvature at $P$ can be defined via limit $$\kappa (P) = \lim_{Q \to P} \sqrt{\frac{24 (s(P,Q) - d(P,Q))}{s(P,Q)^3}},$$ where $s(P,Q)$ is the arc length of the curve $\gamma$ between the points $P$ and $Q$, and $d(P,Q)$ is the length of a line segment from $P$ to $Q$.
Is there a similar geometrical definition of torsion at $P$?
 A: 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 
$$ ds = s(P,Q), dl = d(P,Q). $$

A: (This is not an answer But from the following I believe that torsion can not be obtained from arc length and distance, since torsion is related with curvature)
If $\alpha$ has a unit speed, then we have $$ t:=\alpha',\ n'=-kt
+\tau b,\ b'=-\tau n $$
Consider a surface $$ f(u,v)=u b(v) + \alpha (v) $$ That is unit
normal field is $-n$ so that we have first and second fundamental
forms : $$ E=1,\ F=0, G=1
$$
$$ e=0,\ f=\tau,\ g=k $$
Then we have a Gaussian curvature
$$ K=-\tau^2
$$
(Note that there exists an intrinsic definition of $K$ : $
\lim_{r\rightarrow 0 } 3\frac{2\pi r-C(r)}{\pi r^3} $ where $C(r)$
is a length of geodesic sphere of radius $r$)
Similarly $$  f(u,v)=u n(v) + \alpha (v) $$ Here unit normal field
is $-b$ Then we have a Gaussian curvature $$ K=\frac{-\tau^2
}{(1-k)^2 + \tau^2 }
$$
