---
title: "Curved Mirror Observatory Interactive Lab | Sandbox Physics"
description: "How does mirror shape move the focus and change off-axis image quality? The curvature at the mirror's center sets where rays near the axis focus."
canonical_url: "https://sandboxphysics.com/labs/curved-mirror-observatory/"
language: "en"
page_type: "lab"
alternate_languages:
  zh-CN: "https://sandboxphysics.com/zh/labs/curved-mirror-observatory/"
last_modified: "2026-10-27"
---

Optics 004 · Ray worlds, boundaries, and natural optics

# Curved Mirror Observatory

**How does mirror shape move the focus and change off-axis image quality?** The curvature at the mirror's center sets where rays near the axis focus. A paraboloid sends every ray parallel to the axis to one point. A sphere focuses its outer rays closer to the mirror, so the focal spot smears out. How well the mirror images off-axis points depends on its exact conic shape.

A labeled cutaway of a conic mirror keeps the vertex radius of curvature separate from the clear aperture. The conic mesh, a ruler to the center of curvature, the focus marker, a parallel source and detector samples show each change in space.

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

Reflection from an ideal conic of revolution is computed exactly with geometric rays. Diffraction, central obscuration, coating errors and manufacturing surface errors are not simulated.

Physics tutorial

## Audit the physics of Curved Mirror Observatory

**Background** The mirror is a conic of revolution, defined by its sag: how far the surface dips at each distance from the axis. The vertex radius of curvature and the clear aperture are set independently, and the surface follows $z(r)=\frac{r^2}{R\left(1+\sqrt{1-(1+K)r^2/R^2}\right)}$. Each ray reflects about the normal given by the local slope. This keeps the exact geometric trace separate from the paraxial thin-mirror reference.

**Why it matters** How does mirror shape move the focus and change off-axis image quality?

Start with the essentials

- **Focus question**: How does mirror shape move the focus and change off-axis image quality?
- **One-sentence intuition**: A paraboloid sends rays parallel to its axis to a single focus. On a sphere, where a ray crosses the axis depends on how far from the axis it lands. The detector therefore reports the spot width $\sigma_x$; a single nominal focal length would hide the spread of the marginal rays.

### Core mathematical model

#### Conic sag

$$
z(r)=\frac{r^2}{R\left(1+\sqrt{1-(1+K)r^2/R^2}\right)}
$$

The vertex radius and the conic constant set the drawn surface and its slope at every point.

#### Paraxial focal length of a spherical mirror

$$
f_{\mathrm{paraxial}}=\frac{R}{2}
$$

This value holds for a spherical mirror with a small aperture. Marginal and off-axis rays still have to be traced.

### Common difficulties

#### Applying the mirror equation to every ray

**Typical misconception** All rays from a spherical mirror must cross at the paraxial focal length.

**Better mental model** The mirror equation is a paraxial approximation. Reflecting each ray about the true surface normal shows that the marginal rays cross elsewhere: this is longitudinal spherical aberration.

### Run the experiment

1. 01

   #### Scene 1: Spherical conjugates

   Keep the clear aperture fixed and reduce the vertex radius of curvature; watch the same opening become visibly deeper.

   **What to observe:** With the aperture fixed, a smaller radius of curvature makes the sag larger. On the sphere, the marginal rays also cross the axis closer to the mirror than the paraxial focus.

2. 02

   #### Scene 2: Parabolic focus

   Switch to the parabolic scene without changing the vertex radius.

   **What to observe:** Rays parallel to the axis meet at the parabolic focus to within numerical precision.

3. 03

   #### Scene 3: Off-axis conic spot

   Use the off-axis conic scene and inspect the three-dimensional detector spot.

   **What to observe:** Off-axis light lands on the detector as a spread-out spot, not at a single focus.

Physical conclusion

What a curved mirror does to light follows from its surface slope and where each ray hits it, not from how much the surface looks like a dish.

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)