Spherical Mirrors

Reflection alone gives no focus: a flat mirror sends parallel light away still parallel. Curving the surface gives every point a different normal, and on a sphere that normal is the point's own radius. Take a ray arriving parallel to the axis at height h on a concave mirror of radius R. It strikes where the radius makes an angle θ with the axis, with sin θ = h/R, and because the incoming ray is parallel to the axis, θ is also the angle of incidence. Reflection turns the ray through 2θ, and the triangle made by the centre of curvature C, the strike point and the axis crossing is isosceles — two equal angles θ — which puts the crossing R/(2 cos θ) from C, that is, R − R/(2 cos θ) in front of the vertex. Nothing so far assumes h is small. Now assume it: cos θ → 1 and every ray crosses at R − R/2 = R/2. For R = 40 cm the focus lies 20 cm in front of the vertex, and a ray 6 cm off-axis crosses 2.3 mm short of it.

Now let the light leave a point O on the axis, a distance s in front of the vertex, and follow one ray that meets the mirror at height y. Call α the angle the incoming ray makes with the axis, β the angle of the radius (the normal) and γ the angle of the reflected ray. The angle of incidence is β − α, the angle of reflection is γ − β, and the law of reflection sets them equal, so 2β = α + γ. Paraxially each angle equals its own tangent: α ≈ y/s, β ≈ y/R, γ ≈ y/s′. Substituting and cancelling y gives 2/R = 1/s + 1/s′. That cancellation is the whole point — y has vanished, so every paraxial ray from O arrives at the same s′, which is what makes an image rather than a smear. Writing f = R/2 puts it in working form, 1/s + 1/s′ = 1/f.

Three rays from the object tip are worth drawing because you already know where each goes. A ray parallel to the axis reflects through F. A ray through F reflects parallel to the axis. A ray aimed along the radius, at C, strikes the surface head-on and returns along itself. Any two of them cross at the image tip and the third is your check. A fourth ray strikes the vertex, where the axis is the normal, and reflects at an equal angle on the other side of it; the two similar triangles it makes are where m = −s′/s comes from. On a convex mirror the same three rules hold with F and C behind the surface: the parallel ray reflects as though it had come from F, the ray aimed at F returns parallel, the ray aimed at C returns along itself. Those reflections diverge, so it is their backward extensions that meet — drawn dashed, behind the mirror, giving a virtual image every time.

Measure everything from the vertex, with light arriving from the object side. Then s > 0 for a real object in front; s′ > 0 for an image in front, where rays truly cross and a card catches it; s′ < 0 for an image behind, where only the extensions meet. R and f are positive for a concave mirror, whose C lies in front, and negative for a convex one. Magnification is m = −s′/s, negative when inverted. Run a concave mirror of f = 20 cm through the whole range: at s = 60 cm, s′ = 30 cm and m = −0.50, real, inverted, reduced; at s = 40 cm, which is R, s′ = 40 cm and m = −1.00, so the image lands back on the object; at s = 30 cm, s′ = 60 cm and m = −2.00, real, inverted, magnified; at s = 20 cm the reflected rays leave parallel and there is no image; at s = 10 cm, s′ = −20 cm and m = +2.00, virtual, upright, magnified — the shaving mirror.

A car's passenger-side mirror is convex with R = −2.0 m, so f = −1.0 m. A vehicle 8.0 m back gives 1/s′ = 1/f − 1/s = −1.000 − 0.125 = −1.125 m⁻¹, so s′ = −0.89 m and m = −s′/s = +0.11: an upright virtual image 0.89 m behind the glass at one ninth scale, so a 4.5 m car appears 0.50 m long. Send the object to infinity and s′ → f = −1.0 m; bring it against the glass and s′ → 0. Every image this mirror can form therefore lies inside the 1.0 m of virtual space behind it, upright and reduced, which is exactly why it is fitted — a wide field of view squeezed into a small pane. It is also why the pane carries a warning: the eye reads distance partly from angular size, and this mirror shrinks everything by a factor that itself shrinks with range.

Every step above sent cos θ to 1. Put it back. A ray at height h crosses the axis R − R/(2 cos θ) from the vertex, short of R/2 by (R/2)(sec θ − 1) ≈ h²/(4R). Rays at different heights therefore cross at different points, and the envelope of those crossings is a caustic surface rather than a point. This is spherical aberration, and it is a property of the sphere itself, not a fault in the grinding. It grows fast: the longitudinal miss goes as h² and the transverse blur radius in the paraxial focal plane as about h³/(2R²), so halving the aperture sharpens the image roughly eightfold. Two cures exist. Stop the mirror down and pay in light. Or change the shape — a paraboloid brings all rays parallel to its axis to one exact point, which is why telescope primaries are parabolic, at the cost of blurring everything off-axis.