MIRROR
Same equation as the lens — light forgets whether it bent or bounced.
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
A polished curve and a question as old as still water: where does the reflection live, which way up, and at what size? The answer needs no new law — only the one every flat mirror already obeys, aimed by curvature.
Reflect
θ_in = θ_out at every point; the curve merely tilts the normals so parallel rays fold to a focus at R/2. From there the bookkeeping is identical to the thin lens — light does not remember whether it was bent or bounced.
Solve
The mirror equation hands over dᵢ and the magnification in two lines — and the sign of dᵢ sorts the images into worlds: positive means real and projectable, in front of the glass; negative means virtual, an image you can see but never catch on a screen.
Construct
The classic construction at an experimenter’s pace: the parallel ray reflects through F, the focal ray reflects out parallel, and where they cross — or where their dashed ghosts cross behind the glass — the image stands. Drawn, not assumed.
Audit
Newton’s form closes the books to machine precision, both principal rays are audited onto the image point across the whole object sweep, and reversibility holds: swap object and image and nothing changes. The lens and the mirror are one chapter wearing two costumes.