Ray diagrams for spherical mirrors

To find the image of a point (usually the top of an object standing on the principal axis), draw any two of these four standard rays from that point and see where the reflected rays actually meet, or where their backward extensions meet:

• Ray 1 — parallel to the axis: reflects through the principal focus $F$ (concave mirror), or appears to diverge from $F$ behind a convex mirror.
• Ray 2 — through (or aimed at) $F$: reflects out parallel to the principal axis. Rays 1 and 2 are reverses of each other — the most common and easiest pair to draw.
• Ray 3 — through (or aimed at) the centre of curvature $C$: strikes the mirror along the normal, since a radius meets a sphere perpendicularly, so it reflects straight back on itself.
• Ray 4 — incident at the pole $P$: obeys the ordinary law of reflection, angle of incidence equals angle of reflection, measured from the principal axis acting as the normal at $P$.

Where any two reflected rays actually cross, in front of the mirror, is a real image — solid lines, invertible, catchable on a screen. Where only their backward extensions cross, behind the mirror, is a virtual image — dashed construction lines, since no light truly travels there. For a concave mirror the outcome depends on where the object sits relative to $F$ and $C$; for a convex mirror the reflected rays always diverge, so the image is always virtual, erect and diminished, exactly as the standard result states. Central to this whole scheme is one geometric fact: $F$ always sits at the midpoint of $P$ and $C$, so that

where: $f$ is the focal length (distance $PF$) and $R$ is the radius of curvature (distance $PC$). This relation is why rays 1 and 2 behave as reverses of each other and why the four rays are consistent with one another — it is proved from the geometry of reflection in the derivation below, for rays close to the axis.

The topic is fundamentally a construction technique, but the one formula behind it — $f = R/2$ — deserves a proper geometric proof, since every ray rule quietly assumes it. Consider a ray travelling parallel to the principal axis and striking a concave mirror at a point $M$ close to the pole $P$, so that the ray is paraxial (close to the axis, making only a small angle with it). The line $CM$ is a radius of the mirror, and since a radius meets the spherical surface perpendicularly, $CM$ is the normal at $M$. By the law of reflection, the angle between the incident ray and the normal equals the angle between the reflected ray and the normal; call this angle $\theta$.

Step 1. Because the incident ray is parallel to the axis, and $CM$ is the normal, the angle the incident ray makes with the normal, $\theta$, is also the angle $MCP$ that the radius $CM$ makes with the axis (alternate angles between the parallel ray and $CP$, with $CM$ as transversal).

Step 2. By the law of reflection, the reflected ray also makes angle $\theta$ with the normal $CM$ on the other side. Let this reflected ray cross the axis at point $F$. In triangle $CMF$, the angle at $C$ is $\theta$ (from Step 1) and, since the ray reflects at the same angle to the normal, the angle at $F$ (angle $MFC$, measured from the axis) works out to $2\theta$ — the exterior angle of the triangle equals the sum of the two opposite interior angles, and the angle of incidence equals the angle of reflection at $M$, both $\theta$, giving angle $MFP = 2\theta$.

Step 3. For a paraxial ray, $M$ lies extremely close to $P$, so the strike point is effectively $P$ itself, and triangle $CMF$ becomes, to a very good approximation, isosceles triangle $CPF$ with nearly equal base angles at $C$ and $F$. This forces $PF \approx PC/2$.

Step 4. Reading off the distances: $PF = f$ is the focal length and $PC = R$ is the radius of curvature, so the paraxial construction gives $f = R/2$. This is exactly why the standard ray-diagram rule places $F$ at the midpoint of $P$ and $C$ — it is not an assumption but a consequence of the law of reflection applied to rays close to the axis. The same reasoning, run for a convex mirror, gives $f = R/2$ measured behind the mirror, which is why $F$ there is also midway between $P$ and $C$. This paraxial approximation is exactly why ray diagrams are drawn only with rays close to the axis in practice, even though the sketches exaggerate the angles for clarity.

A concave mirror has a radius of curvature $R = 20\ \text{cm}$, so its focal length is $f = R/2 = 10\ \text{cm}$. An object $AB$ is placed on the axis with its foot at $B$, $15\ \text{cm}$ from the pole — between $F$ (at $10\ \text{cm}$) and $C$ (at $20\ \text{cm}$). Construct the ray diagram and describe the image.

Step 1. Mark $P$, then $F$ at $10\ \text{cm}$ and $C$ at $20\ \text{cm}$ from $P$ along the axis, using $f = R/2 = 10\ \text{cm}$. Draw the object $AB$ as a vertical arrow at $15\ \text{cm}$, tip $A$ above the axis.

Step 2. From $A$, draw Ray 1, parallel to the axis, reflecting through $F$ (by rule, a ray parallel to the axis reflects through the focus).

Step 3. From $A$, draw Ray 2 through $F$: extend the line from $A$ through $F$ to the mirror, then reflect it parallel to the axis (the reverse of Ray 1's rule).

Step 4. The two reflected rays meet beyond $C$, since the object lies between $F$ and $C$ (closer to the mirror than $C$), placing the image farther out than $C$. Call this point $A'$; drop a perpendicular to the axis for $B'$, and join $A'B'$. Because the reflected rays actually converge in front of the mirror, the image is real, inverted, and enlarged, formed beyond $C$ — matching the standard result for an object between $F$ and $C$, and consistent with worked example 1 of the introductory article.

Competitive stretch

If the same mirror's object is moved out to exactly $C$ (at $20\ \text{cm}$), Ray 3 (through $C$) becomes usable: it strikes the mirror along the normal and reflects straight back on itself, unchanged. Combined with Ray 1 (parallel, reflecting through $F$), the two reflected rays now cross exactly at $C$ itself, giving a real, inverted image of the same size as the object — the diagram directly reproduces the well-known "object at C, image at C, same size" result, without needing the mirror formula at all.

• Using rays that are not one of the four standard rays. Only the parallel ray, the focus ray, the centre-of-curvature ray, and the pole ray can be traced reliably by rule.
• Drawing a virtual image's construction lines as solid. Backward extensions used to locate a virtual image must be dashed, since no real light travels behind the mirror.
• Forgetting why the C-ray reflects straight back. It works only because the radius meets the mirror along the normal; bending this ray is a common slip.
• Assuming $f = R/2$ holds for wide, non-paraxial mirrors. The relation is only accurate for rays close to the axis, striking near the pole — the same paraxial condition ray diagrams quietly rely on throughout.

Ask the tutor: If your ray diagram's two reflected rays refuse to meet where the textbook result says they should, ask the AI tutor to check your ray choices and reflection angles step by step.