Where a Lens Forms Its Image
By Lucian — builder & engineer, LK Forge
Move an object toward a converging lens and its image does not just slide along — it flips, grows, races off to infinity, then reappears the right way up and magnified. Every one of those switches happens at a fixed multiple of the focal length, and you can read them straight off one equation. Here is the whole map in a single table, and an interactive ray diagram below so you can watch each row happen.
| Object distance dₒ | Image distance dᵢ | m | Image |
|---|---|---|---|
| Very distant (≫ 2f) | ≈ f (at the focus) | ≈ 0 | Real, inverted, a point |
| Beyond 2f (such as 3f) | 1.5f | −0.5 | Real, inverted, reduced |
| At 2f | 2f | −1 | Real, inverted, same size |
| Between f and 2f (1.5f) | 3f | −2 | Real, inverted, enlarged |
| At f | infinity | — | No image — rays leave parallel |
| Inside f (0.5f) | −1f (same side) | +2 | Virtual, upright, enlarged |
Converging lens (or concave mirror), f > 0. The two turning points are 2f — where the image is the same size — and f, where it escapes to infinity and then flips to a virtual, upright, magnified image: the magnifying-glass regime.
Reproduce any row
Drag the object along the axis in the ray diagram below and watch the three principal rays meet at the image. Set the object beyond 2f for a small inverted image, at 2f for a matched one, inside the focal point for the magnifier — the readout gives dᵢ and m for each position, so you can check them against the table.
Geometric Optics Ray Diagram Simulator — open it full-size, or embed it free on your own site.
Mirrors and diverging elements
A concave mirror obeys the same equation with f = R/2, so its table of cases is identical — the only change is that the real image forms on the same side as the object, in front of the mirror, rather than behind the lens.
A diverging lens and a convex mirror have f < 0, and they have no cases to tabulate: wherever you put the object, the image is always virtual, upright and reduced, sitting between the element and its focal point. That is why a convex mirror makes a good wide-angle security or wing mirror — everything fits, always shrunk and the right way up. Switch the element in the tool above to see it.
For an object at distance dₒ in front of an element of focal length f, solve for the image distance and the magnification:
dᵢ = f·dₒ / (dₒ − f) m = −dᵢ / dₒ
Put in dₒ = 2f and you get dᵢ = 2f, m = −1; put in dₒ = f/2 and you get dᵢ = −f, m = +2. A positive dᵢ is a real image on the far side (a lens) or the near side (a mirror); a negative dᵢ is a virtual image. Every row in the table is one substitution.