Reflection of Light at Spherical Mirrors

The laws of reflection, first met for a plane mirror, apply unchanged at every point of a curved (spherical) mirror as well: the incident ray, the reflected ray and the normal to the surface at the point of incidence all lie in one plane, and the angle of reflection always equals the angle of incidence, both measured from the normal at that point. A spherical mirror is a small, polished section cut from a hollow sphere; if the reflecting surface is the inner (concave) side of the sphere, the mirror is called concave (converging), and if the reflecting surface is the outer (convex) side, the mirror is called convex (diverging). Several fixed points and lines are defined on every spherical mirror: the pole $P$ is the geometric centre of the mirror's reflecting surface; the centre of curvature $C$ is the centre of the sphere of which the mirror is a part; the radius of curvature $R$ is the distance $PC$, i.e. the radius of that sphere; the principal axis is the straight line through $P$ and $C$; and the principal focus $F$ is the point on the principal axis where a narrow beam of rays travelling parallel and close to the principal axis converges (concave mirror) or appears to diverge from (convex mirror) after reflection -- for a spherical mirror of small aperture, $F$ lies midway between $P$ and $C$, so that the focal length $f=PF$ is related to the radius of curvature by the simple paraxial relation $f=\dfrac{R}{2}$. Image formation by a spherical mirror is worked out either by ray-tracing two of a small set of standard rays from a point on the object (a ray parallel to the axis reflecting through/appearing to come from $F$; a ray th …