Optics 008 · Ray worlds, boundaries, and natural optics

Underwater Vision Observatory

A split-level underwater observatory replaces the paraxial cartoon with an exact off-axis meridional image envelope, a real light-versus-projectile aiming experiment, a full critical Snell cone, and a water–glass–air flat-port cutaway. Every colored path has an explicit physical role and shares one refraction model.

Primary boundary result

\mathcal B

$0.50$

Perceptual mapping

\mathcal M

$50\%$

Geometry check

\varepsilon_g

$0.00\pi$

Model regime $\text{valid model regime}$

The interface is a flat mean surface and the water index is fixed at four thirds. The displayed virtual image is the local meridional envelope; surface waves, scattering, Fresnel brightness, polarization, projectile dynamics, and lens distortion are stated omissions.

Physics tutorial

Audit the physics of Underwater Vision Observatory

BackgroundEvery scene starts with tangential optical momentum matching at a flat air–water boundary: $n_{\mathrm w}\sin\theta_{\mathrm w}=n_{\mathrm a}\sin\theta_{\mathrm a}$ . The same relation governs rays leaving a fish, the critical cone seen from below, and each crossing in a parallel water–glass–air camera port. A projectile follows a different mechanical path and does not inherit optical refraction.

Why it mattersHow can one flat water boundary move a fish, mislead a spear, compress the whole sky, and distort a camera port?

Start with the essentials

Focus question: How can one flat water boundary move a fish, mislead a spear, compress the whole sky, and distort a camera port?
One-sentence intuition: The familiar three-quarter-depth rule is only the normal-view limit. The exact local tangential image depth is $d_t=\frac{\left[d^2+\left(1-\mu^2\right)x_s^2\right]^{3/2}}{\mu d^2}$ . It depends on the surface-hit offset, so the rendered neighboring-ray envelope moves continuously instead of staying at an arbitrary fixed scale.

Core mathematical model

Off-axis meridional apparent image

\mu=\frac{n_{\mathrm w}}{n_{\mathrm a}},\qquad d_t=\frac{\left[d^2+\left(1-\mu^2\right)x_s^2\right]^{3/2}}{\mu d^2}

This is the local envelope of neighboring emergent rays in the meridional plane. At normal view it becomes the real depth divided by the relative index; toward the critical angle it approaches the surface. A flat refracting surface is not perfectly stigmatic off axis.

Snell-window critical cone

\theta_c=\arcsin\!\left(\frac{n_{\mathrm a}}{n_{\mathrm w}}\right),\qquad R_{\mathrm{window}}=d\tan\theta_c

The complete above-water hemisphere maps inside this half angle. Outside the circular footprint, an underwater observer receives total-internally-reflected light from below rather than transmitted sky light.

Parallel flat-port shooting equation

p=n_i\sin\theta_i,\qquad \Delta z=\sum_i L_i\frac{p}{\sqrt{n_i^2-p^2}}

One conserved tangential momentum determines the water, glass, and air angles. Solving it for the camera endpoint exposes why a straight pinhole ray crosses a flat port at the wrong position.

Common difficulties

Using one shallow virtual fish for every viewing angle

Typical misconceptionAn underwater object always appears at exactly three quarters of its real depth, so an off-axis ray can be drawn toward that fixed point.

Better mental modelThat ratio is paraxial. The exact neighboring-ray image depends on angle and splits into principal images in three dimensions. This lab renders only the declared meridional envelope and labels it accordingly.

Run the experiment

01
Scene 1: Exact apparent-image envelope
In the apparent-image scene, drag the surface hit from near normal toward the critical side while comparing the real fish, violet envelope, paraxial reference, and exact depth readout.
What to observe: Near normal view, the exact and paraxial depths nearly agree. At larger angle, the exact meridional image rises rapidly and the neighboring backward extensions cease to meet as one perfect point.
02
Scene 2: Aim below the virtual fish
Enter the aiming scene and trace information in the correct direction: photons travel from fish to eye, while the counterfactual spear travels from launcher to water. Compare the naive impact with the gold correction.
What to observe: The straight projectile aimed at the virtual fish crosses the surface without bending and lands beyond the real target. The required below-image correction grows with sight offset and depth.
03
Scene 3: Snell window and flat port
Orbit below the Snell cone, move the selected sky ray, then inspect the side flat-port cutaway. Check the gold horizon rim, magenta total reflection, green two-bend camera rays, and violet pinhole error separately.
What to observe: The horizon maps to the critical rim, not to an arbitrary disc edge. Parallel glass preserves tangential momentum through both port faces, yet the resulting ray family is not the straight single-viewpoint pinhole model.