Conventions for the Lens formula

The diagrams above illustrate the application of the lens formula
\begin{equation} \frac{1}{u}+ \frac{1}{v}=\frac{1}{f} \end{equation}
to  the various possibilities of real and virtual objects and images and the corresponding conventions for the signs of $u$, object distance, $v$, image distance, and $f$, focal length.  The conventions are

• Focal lengths of converging lenses are positive
• Focal lengths of diverging lenses are negative
• Distances of real objects and images from the optical centre are  positive
• Distances of imaginary objects and images from the optical centre are negative. 

The results are for a point object O on the optical axis.  The angle $D$ is the deviation of a ray meeting the lens at a distance $h$ from the optical axis.  Taking the first diagram, we can see that we have taken the angles $\alpha$ and $\beta$ as $h/u$ and $h/v$ respectively, whereas strictly speaking $\tan\alpha=h/u$ and $\tan\beta=h/v$.   In practice however, we are restricted to small angles, as otherwise optical aberrations become significant.  We are then justified in approximating  the tangents of angles by the angles themselves, in which case the deviation may be taken as $D=h/f$, irrespective of the angle at which the ray meets the lens. .