Optics 005 · Ray worlds, boundaries, and natural optics

Caustic Factory

A readable caustic vessel separates incident, reflected, internal, and transmitted segments before highlighting their computed envelope. A radius ruler proves the circular boundary itself changes, while the three scenes explain nephroid reflection, two-interface refraction, and a point-source catacaustic.

Cusp-to-vessel ratio

r_{\mathrm{cusp}}/R_v

$0.50$

Caustic span

\Delta y_{\mathrm{caustic}}

$50\%$

Direction norm error

\Delta_{|\hat{\mathbf k}|}

$0.00\pi$

Model regime $\text{valid model regime}$

Physics tutorial

Audit the physics of Caustic Factory

BackgroundA caustic is the singular envelope of a ray family. It occurs where $\det\!\left(\frac{\partial\mathbf r}{\partial\boldsymbol\xi}\right)=0$ . The vessel control changes the actual circular boundary radius, which is also the curvature radius for a circle. The apparatus keeps incident, reflected, internal, and transmitted segments separately colored.

Why it mattersHow does a smooth family of rays collapse into a brilliant singular envelope?

Start with the essentials

Focus question: How does a smooth family of rays collapse into a brilliant singular envelope?
One-sentence intuition: For parallel rays reflecting inside a circle, the analytic cusp is $r_{\mathrm{cusp}}=R/2$ . The visual glow deliberately remains finite because geometric divergence is regularized by wave effects in any real experiment.

Core mathematical model

Envelope condition

\det\!\left(\frac{\partial\mathbf r}{\partial\boldsymbol\xi}\right)=0

The mapping from ray labels to physical position loses rank at the caustic.

Circular catacaustic cusp

r_{\mathrm{cusp}}=\frac{R}{2}

This analytic benchmark applies to a parallel ray family reflecting from the interior of an ideal circle.

Common difficulties

Reading infinite intensity from bright pixels

Typical misconceptionThe brightest point in the rendered envelope is a quantitative prediction of optical intensity.

Better mental modelThe solver predicts ray density and envelope geometry. Quantitative intensity near the singularity requires wavelength, aperture, coherence, and a wave solver.

Run the experiment

01
Scene 1: Cup nephroid
Keep source angle fixed and change vessel radius; compare the wall, radius ruler, ray-hit positions, and nephroid cusp together.
What to observe: The circular wall and every hit point move with vessel radius; the cusp remains at half that radius while the full envelope rotates with source direction.
02
Scene 2: Refractive glass caustic
Inspect entry, internal, exit, and external segments in the glass-cylinder scene.
What to observe: The central cylinder ray remains undeviated; off-axis rays refract at both interfaces before forming the downstream envelope.
03
Scene 3: Point-source cusp editor
Rotate the interior point source and compare neighboring-ray intersections with the visible cusp.
What to observe: Moving from parallel illumination to an interior point source changes the catacaustic topology without changing the reflection law.