Optics 004 · Ray worlds, boundaries, and natural optics

Curved Mirror Observatory

A labeled cutaway observatory separates vertex curvature radius from clear aperture. A generated conic mesh, center-of-curvature ruler, focus marker, parallel source, and detector samples make every control change spatially visible.

Mean axis crossing

\bar x_f

$0.50$

Longitudinal spread

\sigma_x

$50\%$

Curvature-radius focus

f_{\mathrm{paraxial}}

$0.00\pi$

Model regime $\text{valid model regime}$

Physics tutorial

Audit the physics of Curved Mirror Observatory

BackgroundA rotational conic is defined by its sag rather than by a decorative bowl. Vertex curvature radius and clear aperture are independent; the generated surface uses $z(r)=\frac{r^2}{R\left(1+\sqrt{1-(1+K)r^2/R^2}\right)}$ ; each ray then reflects from the local derivative. This separates the exact geometric trace from the paraxial thin-mirror reference.

Why it mattersHow 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 axial parallel rays to one focus, while a sphere has a radius-dependent crossing. The detector therefore reports $\sigma_x$ instead of hiding marginal-ray spread behind a single nominal focal length.

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 conic constant generate the actual mesh and its local slope.

Paraxial reference

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

This is a small-aperture reference for a spherical mirror, not a replacement for tracing marginal or off-axis rays.

Common difficulties

Applying the mirror equation to every ray

Typical misconceptionAll rays from a spherical mirror must cross at the paraxial focal length.

Better mental modelThe mirror equation is paraxial. Exact surface-normal reflection exposes longitudinal spherical aberration for marginal rays.

Run the experiment

01
Scene 1: Spherical conjugates
Hold clear aperture fixed and reduce vertex curvature radius; watch the same opening become visibly deeper.
What to observe: At fixed aperture, a smaller curvature radius increases sag. Marginal spherical rays also cross closer to the mirror than the paraxial reference.
02
Scene 2: Parabolic focus
Switch to the parabolic scene without changing the vertex radius.
What to observe: Axial parallel rays collapse to the parabolic focus to numerical precision.
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 incidence produces a detector distribution rather than an invented single focus.