# Small deflection approximation

### Curvature

The curvature of a line is given exactly by:

\[
\kappa = \frac{d \phi}{ds}
\]

For small \(\phi\) we note that \(ds \approx dx\), therefore:

\[
\kappa \approx \frac{d \phi}{dx}
\]

### Deflection

\[
\tan \phi = \frac{dv}{dx}
\]

For small deflections, \(\tan \phi \approx \phi\), therefore:

\[
\phi \approx \frac{dv}{dx}
\]