The curvature and torsion: how to distinguish the shape of a curve 
Examples 1
Note that the curves presented below are unique up to a rigid motion of \(\mathbb{R}^{2}\). The examples below also show the variation of the osculating circle along the curve.
1. \(k(t)=t,\; t\in[-18,18]\)
Observation: This curve has an important role in
road construction.
2. \(k(t)=\sin(t),\; t\in[-18,18]\)
3.
\(k(t)=\cos(t),\; t\in[-10,10]\)
4.
\(k(t)=e^{t},\; t\in[-4.5,4.5]\)
5.
\(k(t)=t+\sin(t),\; t\in[-18,18]\)
6.
\(k(t)=t+\cos(t),\; t\in[-18,18]\)
7.
\(k(t)=t+e^{t},\; t\in[-8.5,3.5]\)
8.
\(k(t)=t.\sin(t),\; t\in[-20,20]\)
9.
\(k(t)=t^{2}.\sin(t),\; t\in[-8,8]\)
10. \(k(t)=t.\sin^{2}(t),\; t\in[-20,20]\)
