Move the object along a real-scale bench, change the mirror and see where the exact rays meet.
| Focal length f / radius R | - / - |
| Image distance, mirror equation | - |
| Image distance, exact ray trace | - |
| Difference (trace vs equation) | - |
| Magnification (equation m = −dᵢ/dₒ) | - |
| Magnification (traced tip) | - |
| Image character | - |
| Spot size on screen (RMS) | - |
| Screen | - |
The trace uses every ray you see (law of reflection on the true sphere). With a small aperture the difference is tiny; open the aperture and move the object close to see the image degrade.
| Closed form vs exact trace | - |
| Focus shift at the rim of the aperture | - |
Question: how well does 1/dₒ + 1/dᵢ = 1/f predict where rays really cross? The button traces rays from many object distances and compares.
Geometry. A spherical mirror of radius R has focal length f = R/2 (concave: f > 0, convex: f < 0, plane: f → ∞). Distances are measured from the vertex; real images are in front of the mirror.
Mirror equation. 1/dₒ + 1/dᵢ = 1/f, and the lateral magnification is m = hᵢ/hₒ = −dᵢ/dₒ. Negative m means inverted; |m| > 1 magnified. Real image: dᵢ > 0 (light actually converges, a screen can show it). Virtual image: dᵢ < 0 (rays only appear to diverge from behind the mirror).
Exact ray trace. Every ray is intersected with the true circle (x − R)² + y² = R² and reflected with r = d − 2(d·n)n. Nothing in the trace uses the mirror equation, so comparing the two is an independent check.
Spherical aberration. For rays parallel to the axis at height y the exact crossing point is x = R(1 − 1/(2cosθ)), sinθ = y/R. It moves toward the mirror as y grows; the page compares this closed form with the numerical trace and shows the focus shift at the rim.
Simplifications. Two dimensions (meridional plane only), ideal reflectivity, no diffraction, no light blocked by the object or screen. The blurred image on the screen is drawn from the traced spread of rays (RMS), not from a full wave calculation. Sizes are in centimetres on a bench of about 1.1 m.
Typical values: a bathroom shaving mirror has f of roughly 20–40 cm; convex security mirrors are chosen for a wide field of view at the cost of a reduced image.