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Sandbox Physics

Optics 007 · Ray worlds, boundaries, and natural optics

Fermat Path Finder

An independently initialized three-dimensional apparatus connects Mirror stationarity, Layered-medium route, Continuous-index geodesic. Two dimensional physical controls, direct probe dragging, a detector trace, and three quantitative checks are recalculated from the stated equation.

Interactive modelFermat Path Finder
Primary prediction P1\mathcal P_10.500.50
Physical scale P2\mathcal P_250%50\%
Limit check V\mathcal V0.00π0.00\pi
Model regimevalid model regime\text{valid model regime}

Physics tutorial

How to investigate Fermat Path Finder

BackgroundFermat Path Finder is one independently initialized apparatus with three linked investigations: Mirror stationarity, Layered-medium route, Continuous-index geodesic. Its two controls—Second refractive index and Target offset—feed the governing relation δ ⁣∫ABn ds=0\delta\!\int_A^B n\,\mathrm ds=0. The validity indicator marks the paraxial, lossless, weak-field, or steady-state assumption used by this apparatus.

Why it mattersWhy does a stationary travel-time path reproduce reflection, refraction, and curved rays?

Start with the essentials

Focus question
Why does a stationary travel-time path reproduce reflection, refraction, and curved rays?
One-sentence intuition
The detector curve and all three numerical readouts are recomputed from δ ⁣∫ABn ds=0\delta\!\int_A^B n\,\mathrm ds=0. Geometry and glow are presentation encodings; the equation, units, conservation or limit check, and validity indicator are the quantitative evidence.

Core mathematical model

Governing relation

δ ⁣∫ABn ds=0\delta\!\int_A^B n\,\mathrm ds=0

The implementation evaluates this relation with dimensional inputs and an executable analytic or numerical benchmark. The validity indicator marks the paraxial, lossless, weak-field, or steady-state assumption used by this apparatus.

Common difficulties

Mistaking glow for measured power

Typical misconceptionA brighter cinematic trail must represent proportionally more optical power.

Better mental modelUse the detector and normalized readouts for comparison. Glow is deliberately nonlinear so weak structure stays visible.

Run the experiment

  1. 01

    Scene 1: Mirror stationarity

    Select Mirror stationarity. Sweep Second refractive index, hold Target offset fixed, and then reverse the roles. Drag the stage probe to repeat the first sweep directly.

    What to observe: Read the primary prediction, physical scale, limit check, and validity indicator together. Record where the approximation boundary changes.
  2. 02

    Scene 2: Layered-medium route

    Select Layered-medium route. Sweep Second refractive index, hold Target offset fixed, and then reverse the roles. Drag the stage probe to repeat the first sweep directly.

    What to observe: Read the primary prediction, physical scale, limit check, and validity indicator together. Record where the approximation boundary changes.
  3. 03

    Scene 3: Continuous-index geodesic

    Select Continuous-index geodesic. Sweep Second refractive index, hold Target offset fixed, and then reverse the roles. Drag the stage probe to repeat the first sweep directly.

    What to observe: Read the primary prediction, physical scale, limit check, and validity indicator together. Record where the approximation boundary changes.