---
title: "Mirror Engineering Challenge Interactive Lab | Sandbox Physics"
description: "Can a few mirrors steer a beam to a fixed target just by getting the angles right? Not by angles alone."
canonical_url: "https://sandboxphysics.com/labs/mirror-engineering-challenge/"
language: "en"
page_type: "lab"
alternate_languages:
  zh-CN: "https://sandboxphysics.com/zh/labs/mirror-engineering-challenge/"
last_modified: "2026-10-27"
---

Optics 003 · Ray worlds, boundaries, and natural optics

# Mirror Engineering Challenge

**Can a few mirrors steer a beam to a fixed target just by getting the angles right?** Not by angles alone. Every bounce must land on an actual mirror, and the beam must end on the fixed detector. A ray can meet a mirror's extended line where there is no glass, or be blocked before the target, and either way the design fails.

Route a beam through a periscope, a retroreflector and a mirror maze with obstacles, all with fixed hardware. Exact intersections with finite mirror segments and specular reflection set both the glowing path and the miss distance.

> Interactive simulation: open this page in a web browser to change the parameters and run the model.

This is a geometric ray model with ideal specular mirrors. Diffraction, coating loss, alignment tolerances and finite beam diameter are not included.

Physics tutorial

## Audit the physics of Mirror Engineering Challenge

**Background** Designing a mirror path comes down to finding what each ray hits first. At every mirror hit, the ray leaves along the unit direction $\hat{\mathbf k}_{r}=\hat{\mathbf k}_{i}-2(\hat{\mathbf k}_{i}\cdot\hat{\mathbf n})\hat{\mathbf n}$. That reflection counts only if the ray actually meets the finite mirror segment before it reaches an obstacle or the target.

**Why it matters** Can a few mirrors steer a beam to a fixed target just by getting the angles right?

Start with the essentials

- **Focus question**: Can a few mirrors steer a beam to a fixed target just by getting the angles right?
- **One-sentence intuition**: The target detector stays fixed, and the Lab reports the miss distance $d_{\mathrm{miss}}$. If the target moved to wherever the computed beam ended, every design would look like a success.

### Core mathematical model

#### Direction after a mirror reflection

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

Reflection flips the part of the direction along the mirror normal and keeps the part along the mirror. The direction stays a unit vector.

#### Miss distance from a fixed target

$$
d_{\mathrm{miss}}=\min_{t\ge 0}\left\lVert\mathbf r_0+t\hat{\mathbf k}-\mathbf r_{\mathrm{target}}\right\rVert
$$

The detector ring does not move with the computed ray, so a miss distance of zero means the design really works.

### Common difficulties

#### Adding up angles without checking the hits

**Typical misconception** If the output angle comes out right, the beam must have hit every mirror it was meant to.

**Better mental model** The line a mirror lies on extends forever, but the mirror has edges. A ray can cross that line where there is no mirror, and the Lab rejects such hits before it applies any 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 beam visibly drifts off 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:** In the symmetric reference setup, the beam leaves in exactly the opposite direction to the way it came in. Unequal rotations break this.

3. 03

   #### Scene 3: Occluded ray maze

   Guide the beam through the occluded maze and watch for a ray that misses a finite mirror segment before it reaches the target.

   **What to observe:** When a mirror hit is rejected, the computed path ends there. The Lab does not draw a reflected segment that could not exist.

Physical conclusion

A routing design is solved only when four things agree: the ray directions, the hits on finite mirrors, any blocking, and the miss distance at the fixed target.

Waves & optics · Starter path

## From one ray to interference measurement

Start with refraction, find where the ray picture stops working, then use interference to measure tiny distances.

[Start here → **Optical Boundary Explorer** Turn the incident angle and watch reflected and transmitted power trade off. Then switch to the critical-angle scene. Observe · 11 min](https://sandboxphysics.com/labs/optical-boundary-explorer/)

1. [1 Optical Boundary Explorer](https://sandboxphysics.com/labs/optical-boundary-explorer/)
2. [2 Rainbow Caustic Observatory](https://sandboxphysics.com/labs/rainbow-laboratory/)
3. [3 Camera Obscura](https://sandboxphysics.com/labs/camera-obscura/)
4. [4 Coherent Wave Composer](https://sandboxphysics.com/labs/coherent-wave-composer/)
5. [5 Young Double-Slit Bench](https://sandboxphysics.com/labs/young-double-slit-bench/)
6. [6 Michelson Precision Lab](https://sandboxphysics.com/labs/michelson-precision-lab/)

[View the full path →](https://sandboxphysics.com/topics/waves-and-optics/#starter-path)