Lessons · optics

Image formation by a thin lens: why the ray diagram is the lens equation

Regime 3 — an algebra-only domain, and here the honest second register is geometry, not calculus: the ray diagram. The same image distance falls out of similar triangles, and SymPy proves the three principal-ray constructions agree — which IS the thin-lens equation.

Regime 3 · algebra-only Math machine-derived & checked by SymPy 3 modeling assumptions (author-asserted)

A converging lens with focal length $f = 1$ m. An object $h_o = 0.4$ m tall stands a distance $d_o$ to the left. Slide the object along the axis: outside the focal point it makes a real, inverted image you could catch on a screen (a projector); slide it inside the focal point and the image flips to virtual, upright, and enlarged (a magnifying glass). The thin-lens equation $1/d_o + 1/d_i = 1/f$ tells you where the image is — but the ray diagram shows you why, with three straight lines and no equation at all.

Image distance

d_i = d_o f/(d_o-f)

d_o = 1.5

3 m

Magnification

m = -d_i/d_o

d_o = 1.5

m (a pure number)

-2

Image height

h_i = m\,h_o

d_o = 1.5

h_o = 0.4

-0.8 m

Draw three rays from the tip of the object — no equation needed

\text{(1) parallel}\leftrightarrow F' \qquad \text{(2) through center}\to\text{straight} \qquad \text{(3) } F \leftrightarrow \text{parallel}

Geometric optics needs no calculus and, at first, no algebra. From the object tip, the ray parallel to the axis bends through the far focus $F'$ ; the ray through the lens center goes straight; the ray through the near focus $F$ leaves parallel. Each is a straight line you can rule by hand. Where they cross is the image.

check $d_i = d_o f/(d_o-f)$ solves $1/d_o+1/d_i=1/f$ ; chief ray $m=-d_i/d_o$ ; parallel & focal rays give the same $m$ ; $m = f/(f-d_o)$

✓ The closed form $d_i = \dfrac{d_o f}{d_o - f}$ satisfies the thin-lens equation $\dfrac{1}{d_o} + \dfrac{1}{d_i} = \dfrac{1}{f}$ — back-substituting leaves zero. [structural]
✓ The ray through the lens center is undeviated, so the object and image triangles about the center are similar: $m = h_i/h_o = -d_i/d_o$ . [structural]
✓ The parallel ray refracts toward (or away from) the focus, giving $m = -(d_i - f)/f$ by similar triangles. It equals the chief-ray value $-d_i/d_o$ — the two independent constructions locate the same image point, and that agreement is exactly the thin-lens equation. [structural]
✓ The focal ray (through a focus, emerging parallel) gives $m = -f/(d_o - f)$ — the same magnification again. All three principal-ray constructions locate one image point. [structural]
✓ The magnification in terms of $d_o, f$ alone: $m = -\dfrac{d_i}{d_o} = \dfrac{f}{f - d_o}$ — its sign sets the orientation (upright when $m>0$ , inverted when $m<0$ ), its size the scale. [structural]

Dimensional homogeneity: checked by SymPy (holds).

✗ Common misconception: “A taller object is magnified more — the magnification depends on the object's size.”

It does not. The magnification $m = -d_i/d_o$ is fixed by the two distances alone — a property of the lens and where you place the object, not of the object's height. Drag the object-height slider: the image grows and shrinks in step with the object, but $m$ (and whether the image is enlarged or reduced) never changes. Magnification is about the geometry of the rays, which is why the same lens magnifies a tall tree and a short flower by the same factor at the same distance.

Modeling assumptions — author-asserted, disclosed not discharged

A thin lens: its thickness is negligible next to the focal length, so both refractions happen in one plane at the lens center.
Paraxial rays — all rays stay close to the axis, so a point object maps to a single point image. Real lenses with wide apertures blur this (spherical aberration).
A single converging lens, $f > 0$ , with the sign convention that a real image (on the far side, catchable on a screen) has $d_i > 0$ and a virtual image has $d_i < 0$ .

The ray diagram, fully annotated

A static rendering (Matplotlib) at the default object distance — the three principal rays converging to the real, inverted image. The interactive version, with object-distance and height sliders, is in the Graph tab above.

Formulas used

Hover a formula to preview its reference entry; click to open it in the reference (or the concept graph):