Mirrors — plane, concave, convex; mirror equation 1 - v + 1 - u = 1 - f: Foundations

This page assumes nothing. Before you can read a single line of the parent topic, you need a small toolkit of ideas. We build each one from a picture, say what it looks like, and say why the topic can't live without it. Read top to bottom — each tool leans on the one above it.

1. A ray of light — the arrow we track

The picture: a straight line with an arrow. Nothing more.

Why the topic needs it: every mirror argument is "an arrow comes in, an arrow leaves." Without a ray to draw, there is nothing to reflect.

Figure — Mirrors — plane, concave, convex; mirror equation 1 - v + 1 - u = 1 - f

2. The normal — the "straight-out" line

The picture: a dashed line making a perfect square-corner with the surface at the hit-point (look at the dashed line in the next figure).

Why the topic needs it: the law of reflection measures angles from the normal, not from the mirror. And for a curved mirror the normal at any point aims straight at the sphere's centre — that single fact powers the $f = R /2$ derivation.

3. Angle of incidence and angle of reflection

The picture: two wedges opening away from the dashed normal — one for the arriving arrow, one for the leaving arrow.

Why the topic needs it: this is the only physical law in the whole chapter. Everything else is geometry stacked on top of $i = r$ . See Reflection of Light — Laws & Normal for a deeper drill.

4. The principal axis, pole, centre & radius of curvature

A curved mirror is a shiny slice cut from a hollow ball (a sphere). To talk about it we name a few landmarks.

The picture: a curved arc (the mirror), a dot $C$ off to one side, a straight horizontal line joining $C$ through the middle-dot $P$ . Every "spoke" from $C$ to the arc is a radius, and — crucially — each spoke is a normal (it hits the arc square-on).

Why the topic needs it: these names are the vocabulary of every ray diagram. The $f = R /2$ proof is literally "draw the radius (=normal), apply $i = r$ , spot an isosceles triangle."

5. Focus ( $F$ ) and focal length ( $f$ )

The picture: several parallel arrows hit a concave mirror and squeeze together to one bright dot — that dot is $F$ .

Why the topic needs it: $F$ is where the mirror's "converging power" lives, and $f$ is the number that goes into the mirror equation. The parent page proves $f = R /2$ , i.e. the focus sits halfway between the mirror and the ball's centre.

6. Similar triangles — the geometry engine

The picture: a small triangle and a big triangle with identical corner angles — one is a scaled-up photocopy of the other.

$\frac{side _{1}^{small}}{side _{1}^{big}} = \frac{side _{2}^{small}}{side _{2}^{big}}$

7. Signed distances — measuring with direction

The symbols this creates:

Symbol	Plain meaning	Picture	Sign for a real object
$u$	object distance ( $P$ →object)	arrow left of mirror	$u < 0$
$v$	image distance ( $P$ →image)	arrow to image	$-$ if front, $+$ if behind
$f$	focal length ( $P$ → $F$ )	$P$ to focus dot	concave $-$ , convex $+$
$R$	radius ( $P$ → $C$ )	$P$ to ball-centre	concave $-$ , convex $+$
$h$	object height	upright arrow	$+$ (up)
$h^{'}$	image height	image arrow	$-$ if inverted

Why the topic needs it: the mirror equation $\frac{1}{v} + \frac{1}{u} = \frac{1}{f}$ and the magnification $m = - \frac{v}{u}$ are only true with signed numbers. Feed in raw magnitudes and you'll predict images that don't exist.

A straight arrow showing the path and direction light travels.

Where is the normal drawn, and what angle does it make with the surface?

At the hit-point, perpendicular (

9 0^{\circ}

) to the surface.

State the law of reflection.

The angle of incidence equals the angle of reflection (

i = r

), both measured from the normal.

On a spherical mirror, which line is always the normal at a point?

The radius — the line from that point back to the centre of curvature

C

Name the pole, centre of curvature, and radius in one sentence each.

Pole

P

= middle of the surface; centre

C

= centre of the sphere; radius

R = P C

What defines the focus

F

The point where rays parallel to the principal axis meet (or appear to come from) after reflecting.

When are two triangles "similar," and what follows?

When their angles match; then matching sides share one common ratio.

Under the sign convention, what sign is a real object's distance

u

Negative (

u < 0

), because it lies against the incident light.

Write magnification two ways.

m = \frac{h ^{'}}{h} = - \frac{v}{u}

Why does the mirror equation use

\frac{1}{v}, \frac{1}{u}, \frac{1}{f}

instead of

v, u, f

Dividing the derived relation

uv = v f + u f

uv f

leaves reciprocals as the tidy result.

Foundations — Mirrors — plane, concave, convex; mirror equation 1 - v + 1 - u = 1 - f

1. A ray of light — the arrow we track

2. The normal — the "straight-out" line

3. Angle of incidence and angle of reflection

4. The principal axis, pole, centre & radius of curvature

5. Focus ( $F$ ) and focal length ( $f$ )

6. Similar triangles — the geometry engine

7. Signed distances — measuring with direction

8. Magnification ( $m$ ) — the size/orientation report

9. Reciprocals — why the equation has $\frac{1}{v}$ , not $v$

Prerequisite map

Equipment checklist