If a curve $C$ is defined by

 $\mathbf{R}\=\left(a\+{\int }_c^s\cos \left(\mathrm{\θ}\left(u\right)\right) \mathit{\ⅆ}u\right) \mathbf{i}\+\left(b\+{\int }_c^s\sin \left(\mathrm{\θ}\left(u\right)\right) \mathit{\ⅆ}u\right) \mathbf{j}$ 

then

$\mathbf{R}\prime \=\mathbf{T}\=\cos \left(\mathrm{\θ}\left(s\right)\right) \mathbf{i}\+\sin \left(\mathrm{\θ}\left(s\right)\right) \mathbf{j}$

and clearly, $\parallel \mathbf{R}\prime \parallel \=1$. Also,

$\mathbf{T}\prime \= \left(-\sin \left(\mathrm{\θ}\left(s\right)\right) \mathbf{i}\+\cos \left(\mathrm{\θ}\left(s\right)\right) \mathbf{j}\right) \mathrm{\θ}\prime \left(s\right)$ 

where $\mathrm{\θ}\left(s\right)\={\int }_c^s\mathrm{\κ}\left(t\right) \mathrm{dt}$, and $\mathrm{\θ}\prime \left(s\right)\=\mathrm{\κ}\left(s\right)$ by the Fundamental Theorem of Calculus.

Consequently, $\parallel \mathbf{T}\prime \parallel \=\mathrm{\θ}\prime \left(s\right)\=\mathrm{\κ}\left(s\right)$.