Osculating helix at any point of a space curve

EDIT1:

A 3d curve passes through the origin, the $$(T,N,B)$$ are tangent, normal and binormal vectors respectively. In the (T,N) spanned plane of osculation can we parametrize a helix of osculation (with given curvature $$\kappa$$ and torsion $$\tau$$) as

$$(x,y,z)= \left(\frac{\kappa}{\kappa^2+\tau^2}\sin t,\, \frac{\tau}{\kappa^2+\tau^2}(1-\cos t),\,\frac{t \tau }{2\pi(\kappa^2+\tau^2)}\right)\,$$

(where $$t$$ is rotation in the projected osculation plane around vector B ) ? If not, how is it correctly parametrized? Thanks in advance.

• Just to confirm the terminology of helix of osculation: The request is for the unit-speed helix of constant curvature $\kappa$ and torsion $\tau$? Commented Apr 2, 2022 at 23:30
• What does it mean when you say “in the $(T,N)$ plane”. Commented Apr 3, 2022 at 4:05
• Have you calculated the curvature and torsion of this curve? Commented Apr 3, 2022 at 4:42
• Yes, they are $(r,p)/(r^2+p^2)$ respy, where r is the helix radius and pitch $p=2 \pi r$ Commented Apr 3, 2022 at 6:23
• @AndrewD.Hwang I thought we can take unit speed or any other parametrization. Is it right? Commented Apr 3, 2022 at 6:33