Relation Between f and R

For a concave mirror, a paraxial ray parallel to the principal axis strikes the mirror at $M$ and reflects through the principal focus $F$; the line $CM$ (through the centre of curvature) is the normal at $M$, and the angle of incidence there equals the angle of reflection, both denoted $i$. Dropping the perpendicular $MP$ from $M$ onto the axis, the right triangles $\triangle MCP$ and $\triangle MFP$ give $\tan i = PM/PC$ and $\tan 2i = PM/PF$ (using the exterior-angle property that $\angle MFP=2i$). For paraxial rays, $\tan i\approx i$ and $\tan 2i\approx 2i$, so $i=PM/PC$ and $2i=PM/PF$; dividing these two relations gives $PF=PC/2$, i.e. $\boxed{f=R/2}$. The identical construction, wo …