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Sandbox Physics

Optics 003 · Ray worlds, boundaries, and natural optics

Mirror Engineering Challenge

A beam-routing bay tests a periscope, retroreflector, and occluded mirror maze against fixed hardware. Exact finite-segment intersections and specular reflections drive both the luminous path and the miss-distance readout.

Interactive modelMirror Engineering Challenge
Output direction θout\theta_{\mathrm{out}}0.500.50
Fixed-target miss dmissd_{\mathrm{miss}}50%50\%
Direction norm error Δ∣k^∣\Delta_{|\hat{\mathbf k}|}0.00π0.00\pi
Model regimevalid model regime\text{valid model regime}

Physics tutorial

Audit the physics of Mirror Engineering Challenge

BackgroundMirror engineering is a constrained intersection problem. At every hit, the outgoing unit direction is k^r=k^i−2(k^i⋅n^)n^\hat{\mathbf k}_{r}=\hat{\mathbf k}_{i}-2(\hat{\mathbf k}_{i}\cdot\hat{\mathbf n})\hat{\mathbf n}, but that reflection is valid only when the incoming ray actually intersects the finite mirror segment before an obstacle or target.

Why it mattersCan a compact mirror network steer a beam without losing field of view?

Start with the essentials

Focus question
Can a compact mirror network steer a beam without losing field of view?
One-sentence intuition
A convincing challenge keeps the detector fixed and reports dmissd_{\mathrm{miss}}. Moving the target to the computed endpoint would turn every design into a false success.

Core mathematical model

Specular direction update

k^r=k^i−2(k^i⋅n^)n^\hat{\mathbf k}_{r}=\hat{\mathbf k}_{i}-2(\hat{\mathbf k}_{i}\cdot\hat{\mathbf n})\hat{\mathbf n}

The reflection preserves direction norm and reverses the normal component while retaining the tangential component.

Fixed-target miss distance

dmiss=min⁡t≥0∥r0+tk^−rtarget∥d_{\mathrm{miss}}=\min_{t\ge 0}\left\lVert\mathbf r_0+t\hat{\mathbf k}-\mathbf r_{\mathrm{target}}\right\rVert

The detector ring remains independent of the ray solve, making zero miss a meaningful design condition.

Common difficulties

Using angle sums without hit tests

Typical misconceptionIf the output angle formula is correct, the beam must have touched every intended mirror.

Better mental modelAn infinite supporting line can intersect where the finite mirror does not exist. The apparatus rejects those hits before applying reflection.

Run the experiment

  1. 01

    Scene 1: Periscope build

    Start from the aligned periscope and rotate one mirror away from its reference angle.

    What to observe: The luminous beam visibly leaves the fixed target and the miss-distance readout rises.
  2. 02

    Scene 2: Corner retroreflector

    Test the retroreflector with equal and unequal mirror rotations.

    What to observe: The symmetric reference returns the direction antiparallel to the input; asymmetric rotation breaks that condition.
  3. 03

    Scene 3: Occluded ray maze

    Route through the occluded maze and watch for a missing finite-segment hit before the target.

    What to observe: A rejected mirror hit terminates the computed route rather than drawing an impossible reflected segment.