Uses of convex mirrors

A convex mirror's curved, outward-bulging surface reflects light rays that would have missed a flat mirror entirely, bending them back toward the viewer's eye. The result is that a convex mirror always shows an image that is erect and diminished — smaller than the real object — regardless of how close or far the object is. This diminishment can be described by the mirror's magnification, the ratio of image height to object height:

where: $h_i$ is the height of the image, $h_o$ is the height of the real object, and $m$ is always positive (the image is erect, not inverted) and always less than one (the image is diminished, never magnified). This is exactly what makes a convex mirror's field of view so useful.

Field of view and curvature. A more curved mirror — one with a smaller radius of curvature $R$ — bends light through a wider range of directions than a gently curved one, so it gathers rays from a wider slice of the scene in front of it. For a mirror of width $w$ that is small compared to its radius of curvature, the angular field of view $\omega$ it captures follows roughly:

where: $w$ is the width of the mirror, $R$ is its radius of curvature (a flatter mirror has a larger $R$; a more sharply bulged mirror has a smaller $R$), and $\omega$ is the angular field of view the mirror captures. A perfectly flat mirror is the limit $R \to \infty$, giving a narrow field of view for its size; shrinking $R$ (bulging the mirror more) widens $\omega$ for the very same mirror width — the reason road-bend and shop-security mirrors are often noticeably more curved than a vehicle's flatter side mirror. Three everyday devices exploit this trade-off:

• Vehicle side-view (and rear-view) mirrors: a convex mirror lets a driver see a much wider strip of road behind and beside the vehicle than a flat mirror of the same size, catching vehicles that would otherwise sit in a blind spot. Because everything looks smaller, it also looks farther away — the reason for the warning printed on the glass.
• Road-safety mirrors at blind bends: at sharp turns, especially on hilly or narrow roads, a large, strongly curved convex mirror lets drivers approaching from either direction glimpse a wide arc of the road around the corner.
• Shop and corridor surveillance mirrors: a single large convex mirror mounted in a corner lets one person watch a wide area of a shop floor, replacing several flat mirrors and closing off blind spots.

The trade-off is not a coincidence — it follows directly from how a curved surface reflects light. Every point on a flat mirror reflects light in only one fixed direction relative to the incoming ray, so a flat mirror of a given width only "sees" a narrow cone of the world lying roughly straight in front of it. A convex mirror's surface, by contrast, is tilted at a different angle at every point across its width, because it curves away from the viewer. Light arriving from well off to the side still strikes some part of the bulging surface at just the right angle to bounce back to the eye. The more sharply the surface curves (the smaller its radius of curvature $R$), the more steeply the surface tilts from edge to edge, so rays from an even wider spread of directions get redirected inward — hence the relation $\omega \approx w/R$: shrink $R$ and the same width $w$ gathers a wider angle $\omega$. This same curvature is exactly what shrinks the image. A curved surface effectively "compresses" a wide scene onto a small patch of glass — each point on the mirror stands in for a wider slice of the real world than the equivalent point on a flat mirror would. Squeezing more of the world onto the same-sized image necessarily makes every object in it look smaller, which is precisely the diminished image described by equation (1). Wide field of view and diminished image size are therefore two faces of the same geometric fact, not two separate properties — a designer cannot choose one without accepting the other, which is exactly why every use of a convex mirror accepts smaller-looking objects as the price of a wider view.

Question: Traffic engineers want to install a convex mirror of width $w = 0.40\ \text{m}$ at a blind hillside bend. To let a driver see both directions of the curving road at once, the mirror must provide a field of view of at least $50^\circ$. Using $\omega \approx w/R$, find the largest radius of curvature $R$ the mirror can have, and say what that means about how strongly curved the mirror must be.

Step 1. Convert the required field of view to radians, since equation (2) needs $\omega$ in radians: $\omega = 50^\circ \times \dfrac{\pi}{180^\circ} \approx 0.87\ \text{rad}$.

Step 2. Rearrange $\omega \approx w/R$ for $R$: $R \approx \dfrac{w}{\omega} = \dfrac{0.40\ \text{m}}{0.87} \approx 0.46\ \text{m}$.

Step 3. Interpret the number. A radius of curvature of only about $0.46\ \text{m}$ describes a mirror that bulges quite sharply — nothing like the gently curved mirror on a car door. This matches what is actually seen on hillside roads: the bend mirrors are noticeably more rounded than a vehicle's side mirror, precisely because they must squeeze a much wider field of view out of a similarly modest-sized disc of glass. The mirror needs a radius of curvature of roughly $0.46\ \text{m}$ or smaller — a strongly curved convex mirror — to meet the required field of view.

Competitive stretch

If the same engineers doubled the mirror's width to $w = 0.80\ \text{m}$ while keeping $R \approx 0.46\ \text{m}$, equation (2) gives $\omega \approx 0.80/0.46 \approx 1.74\ \text{rad} \approx 100^\circ$ — roughly double the field of view for the same curvature, since $\omega$ scales directly with $w$ at fixed $R$. This shows engineers have two independent knobs — mirror size and curvature — and either one alone can widen the field of view, though a bigger mirror costs more glass while a smaller $R$ shrinks the image further.

• Thinking a wide field of view is "free" — it always comes with the image looking smaller, and therefore farther away, than it truly is.
• Misreading the "objects in mirror are closer than they appear" warning as a manufacturing flaw, when it is a direct, predictable consequence of the diminished image described by equation (1).
• Assuming a convex mirror could replace a concave mirror for close-up magnified viewing, such as a dentist's mirror or a shaving mirror — a convex mirror's magnification $m$ is always less than one, so it can never magnify.
• Forgetting that smaller radius of curvature means a more curved, wider-viewing mirror, not a flatter one — it is easy to muddle "smaller $R$" with "less curved."

Ask the tutor: Want to work through more field-of-view or magnification numbers for a real vehicle mirror or shop layout?