Reflection from curved mirrors

Curved (spherical) mirrors -- concave (polished from the inside) or convex (polished from the outside) -- are needed whenever a parallel or divergent beam of light must be brought to a focus, since a flat mirror alone cannot do this; familiar examples include torch and headlight reflectors, vehicle rear-view mirrors, and (for the very sharpest focusing) parabolic mirrors used in searchlights and reflecting telescopes.

The RADIUS OF CURVATURE $R$ of a spherical mirror is the radius of the sphere it is cut from. For spherical mirrors specifically, the FOCAL LENGTH is exactly half the radius of curvature, $f = \dfrac{R}{2}$: for a concave mirror this is the distance at which parallel incident rays actually converge after reflection; for a convex mirror it is the distance BEHIND the mirror from which the reflected rays appear to be diverging. By the sign convention (incident rays travelling left to right, facing the polished surface), the focal length of a convex mirror comes out POSITIVE, while that of a concave mirror comes out NEGATIVE.

For a small mirror (aperture at least ten times smaller than $u$, $v$ and $f$), object distance $u$, image distance $v$ and focal length $f$ are related by the MIRROR FORMULA, $\dfrac{1}{f} = \dfrac{1}{v}+\dfrac{1}{u}$. The converging or diverging ABILITY of a mirror (or lens) is its FOCAL POWER, $P = \dfrac{1}{f}$, measured in dioptres (D) in SI units. The LATERAL MAGNIFICATION -- the ratio of image size to object size, measured perpendicular to the axis -- is $m = -\dfrac{v}{u}$.

For ANY position of the object, a convex mirror always forms a virtual, erect, diminished image, $|m|<1$. For a CONCAVE mirror, the outcome depends entirely on where the object sits relative to $f$ and $2f (=R)$: at $u=\infty$ the image forms exactly at the focus, point-sized ($m=0$); for $u$ anywhere between $2f$ and infinity, the image is real, inverted and diminished, forming between $f$ and $2f$ ($|m|<1$); at $u=2f$ exactly, the image is real, inverted, same size, also at $2f$ ($m=-1$); for $u$ between $f$ and $2f$, the image is real, inverted and MAGNIFIED, forming beyond $2f$ ($|m|>1$); at $u=f$ exactly, the image forms at infinity; and for $u$ less than $f$ (object placed inside the focus), the image is virtual, erect and magnified, forming behind the mirror ($|m|>1$).

Worked illustration: a 20 cm long pencil lies along the axis of a concave mirror of $R=30$ cm (so $f=-15$ cm), with its near end 20 cm from the pole. For the near end, $u_1=-20$ cm: $\frac{1}{-15}=\frac{1}{v_1}+\frac{1}{-20}\Rightarrow v_1=-60$ cm. For the far end, $u_2=-40$ cm (20 cm + 20 cm pencil length): $\frac{1}{-15}=\frac{1}{v_2}+\frac{1}{-40}\Rightarrow v_2=-24$ cm. So the image runs from $-60$ cm to $-24$ cm, giving an image length of $60-24=36$ cm. …