Curved mirror terms

Before you can draw a single ray diagram for a concave or convex mirror, you need the vocabulary that labels its parts — the way you need left and right before giving directions. A spherical mirror is really just a small curved patch cut from a hollow sphere, and every point or line marked on it — the pole, the centre of curvature, the principal axis, the focus — refers back to that parent sphere. Once these six terms are fixed in memory, ray diagrams stop being a memorised drawing and start being a picture you can reconstruct from first principles: draw the sphere in your head, mark its centre, and everything else falls into place. This article gathers the standard definitions, shows geometrically why the focus always sits near the midpoint between pole and centre, and works two labelling problems end to end.

The core insight

Every spherical mirror is defined by its pole, centre of curvature, principal axis, and principal focus, with focal length always half the radius of curvature, f=R/2f = R/2, because the focus is geometrically the midpoint of the segment joining pole and centre for rays close to the axis.

The core idea

A spherical mirror is a small cap sliced off a hollow glass or metal sphere and silvered on one side. Everything you name on the mirror is a reference back to that parent sphere.

  • Pole (P): the geometric centre of the mirror's reflecting surface — the "middle point" of the cap.
  • Centre of curvature (C): the centre of the sphere the mirror belongs to. For a concave mirror C sits in front, on the reflecting side; for a convex mirror C sits behind the mirror, on the far side from the object.
  • Radius of curvature (R): the distance from P to C — literally the radius of the parent sphere.
  • Principal axis: the straight line through P and C, extended both ways. It is the fixed reference line every ray diagram is built around.
  • Principal focus (F): the point on the principal axis where rays travelling parallel to the axis meet after reflection (concave), or the point they appear to spread out from after reflection (convex).
  • Focal length (f): the distance from P to F along the principal axis.

Two mirrors, two placements. In a concave mirror the reflecting surface curves inward like the inside of a bowl, so C and F both lie in front of the mirror, on the same side as the object and the incoming light. In a convex mirror the reflecting surface bulges outward, so C and F both lie behind the mirror — the reflected rays only appear to come from there, since nothing physically reaches that point. These six terms are linked by one stated relationship:

f=R2f = \frac{R}{2}

where: ff is the focal length, RR is the radius of curvature, and both are measured along the principal axis from the pole. The relation says the focus is always the midpoint of PC, whichever type of mirror you are looking at.

Figure
Labelled diagram of a concave mirror showing pole centre of curvature focus and principal axis with focal length half the radius
Concave mirror — pole, focus and centre of curvature on the principal axis principal axis P F C f (focal length) R (radius of curvature) ray parallel to axis F is the midpoint of PC, so f = R / 2

Why the focus sits at the midpoint

The relation f=R/2f = R/2 is a stated fact at this level, but it is worth seeing where it comes from, because the geometry behind it is exactly what makes ray diagrams work and it uses nothing beyond the reflection law and a triangle you can draw with a ruler.

Derivation AWhy the principal focus is the midpoint of the pole and the centre of curvature

The goal is to show, using only the law of reflection and the geometry of a circle, that for a ray landing close to the pole a concave mirror sends any axis-parallel ray through a point that sits almost exactly halfway between the pole and the centre of curvature, which is the geometric reason the stated relationship f=R/2f = R/2 holds and why it is only reliable for rays that stay close to the principal axis.

Step 1. Pick a point M on the mirror's surface, close to the pole P, and let a ray travel in parallel to the principal axis and strike the mirror at M. Because M lies on the sphere centred at C, the line from C to M is a normal to the mirror at that point — it is a basic property of a sphere that every radius meets the surface at right angles.

Step 2. Apply the law of reflection at M: the angle between the incoming ray and the normal CM equals the angle between the reflected ray and CM. Call this common angle θ\theta. Since the incoming ray is parallel to the principal axis, the angle the normal CM makes with the axis at C, namely angle PCM, also equals θ\theta — the two angles are related by the parallel lines, exactly as alternate or corresponding angles are in any diagram with a transversal cutting parallel lines.

Step 3. The reflected ray leaves M and crosses the principal axis at some point F. In the triangle formed by C, M and F, the angle at C is θ\theta and the angle at M (between MF and MC) is also θ\theta, by the reflection law. A triangle with two equal angles is isosceles, so the two sides opposite those angles are equal: CF=MFCF = MF.

Step 4. Now use the paraxial condition — M is taken close to the pole P, which is the working assumption behind every ray diagram at this level. As M approaches P, the length MF approaches PF. Combined with Step 3, this gives CF≈PFCF \approx PF, meaning F is (approximately) the midpoint of segment PC. Since PC=RPC = R and F splits it in half, PF=R/2PF = R/2, which is exactly the stated relationship f=R/2f = R/2. Check: the approximation only holds for rays close to the axis — a ray striking near the mirror's edge crosses the axis at a point noticeably short of this midpoint, which is why real mirrors used for sharp images are made with a small aperture relative to R.

Worked example

A textbook diagram shows a concave mirror with its centre of curvature marked 20 cm from the pole along the principal axis. A second, convex mirror in the same set has a centre of curvature 16 cm from its pole. Find the focal length of each and state where the focus lies relative to the mirror.

Step 1. For the concave mirror, the distance from pole to centre of curvature is the radius of curvature, so R=20 cmR = 20\ \text{cm}.

Step 2. Apply f=R/2f = R/2: f=20/2=10 cmf = 20/2 = 10\ \text{cm}. Because it is concave, both C and F lie in front of the mirror, on the principal axis, with F at 10 cm from the pole — exactly midway to C.

Step 3. For the convex mirror, R=16 cmR = 16\ \text{cm}, so f=16/2=8 cmf = 16/2 = 8\ \text{cm}. Because it is convex, both C and F lie behind the mirror, with F at 8 cm behind the pole, again midway to C.

Step 4. Compare the two: a smaller radius of curvature gives a smaller focal length in both cases — the relationship f=R/2f = R/2 holds regardless of mirror type, only the side on which C and F sit changes.

The concave mirror has focal length 10 cm with F in front of the mirror; the convex mirror has focal length 8 cm with F behind the mirror.

Common traps

  • Confusing centre of curvature (C) with focus (F) — C is the centre of the entire parent sphere, while F is the point midway between P and C.
  • Forgetting that a convex mirror's C and F both sit behind the mirror, not in front of it, even though the mirror still "faces" the object.
  • Treating radius of curvature (PC) and focal length (PF) as the same measurement — they are related by f=R/2f = R/2 but are two different distances.
  • Applying the geometric picture with rays far from the axis and expecting them to still cross exactly at F — the midpoint result only holds for rays close to the principal axis.

Ask the tutor: Mixing up pole, centre of curvature and focus, or want a quick drill on which side of a convex mirror these points sit on — ask the AI tutor to quiz you until it sticks.

Why does the focus end up exactly halfway between the pole and the centre of curvature?

It comes from the law of reflection applied to a ray close to the axis. The normal at any point on the mirror passes through C, and reflection makes the ray's angle to that normal equal on both sides, which forces the triangle formed by the centre, the strike point, and the point where the ray crosses the axis to be isosceles. For a strike point near the pole, that isosceles condition collapses to the focus sitting almost exactly midway between P and C, giving f=R/2f = R/2.

Does f = R/2 apply to both concave and convex mirrors the same way?

Yes, the relationship between the sizes of f and R is identical for both — only the side of the mirror on which C and F actually sit changes. For a concave mirror both are in front, on the reflecting side; for a convex mirror both are behind. The magnitude relationship, focal length equal to half the radius of curvature, does not depend on which type of mirror you have.

Why can I not just plug these terms into a formula like 1/v + 1/u = 1/f right away?

That formula, the mirror formula, connects object distance, image distance and focal length numerically, and it belongs to the next level of study, not this stage. At this stage the goal is only to define and correctly locate pole, centre of curvature, radius of curvature, principal axis, focus and focal length on a diagram, and to know the stated relationship f=R/2f = R/2. Numerical image-position problems using the mirror formula come later.

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