LENS
Three rays, one meeting point — geometry and algebra agree to the last digit.
Use the simulation above to change the variables and play through the guided stages. The explanation below describes the default starting values; the simulation updates its explanation as you experiment.
Setup
An object, a converging lens, a focal length. Somewhere — maybe behind the lens, maybe impossibly in front of it — sits an image. Geometry alone will find it, using nothing but three straight lines.
Rules
Three rays with memorable manners: the parallel ray bends through the far focus, the central ray doesn’t bend at all, the focal ray leaves parallel. Any two of them locate the image; the third tags along as a built-in consistency check.
Solve
The algebra that matches the drawing: 1/dₒ + 1/dᵢ = 1/f. Solve it and the sign does the interpreting — positive dᵢ is a real image you could catch on paper; negative is a virtual one that lives only in the looking.
Trace
Watch the construction draw itself, one ray at a time. No equation is consulted — just three rules and a straightedge — and yet all three lines arrive at a single point. When geometry and algebra agree this well, one of them is redundant. Keep both anyway.
Audit
The ledger: the lens equation to machine precision, Newton’s form (dₒ−f)(dᵢ−f) = f² as a second witness, and every ray individually confirmed to pass through the computed image point. A camera, a projector, and a magnifying glass are this page with different numbers.