Optics 008 · Ray worlds, boundaries, and natural optics

Underwater Vision Observatory

How can one flat water surface make a fish look shallower, fool a spear, squeeze the sky and distort photos? All four are refraction at the same flat surface. It lifts the fish's image toward the surface, more so when you look at a slant, so a spear thrown straight at that image passes over the fish. From below, the whole sky is squeezed into a cone about 49 degrees from vertical, set by the critical angle. A flat camera port bends light the same way and distorts the picture.

A split view above and below the water shows the exact off-axis meridional image envelope, an aiming experiment that compares light with a projectile, the full Snell cone at the critical angle and a cutaway of a flat water–glass–air port. Every colored path has a defined physical role, and all of them share 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 water surface, and the water index is fixed at four thirds. The virtual image shown is the local meridional envelope. Surface waves, scattering, Fresnel brightness, polarization, projectile dynamics and lens distortion are not included.

Physics tutorial

Audit the physics of Underwater Vision Observatory

BackgroundAll three scenes start from one rule at a flat air–water boundary: the tangential optical momentum matches on both sides, $n_{\mathrm w}\sin\theta_{\mathrm w}=n_{\mathrm a}\sin\theta_{\mathrm a}$ . This relation governs the rays leaving a fish, the critical cone seen from below, and every crossing in a camera port made of parallel water, glass and air layers. A projectile is different: it follows a mechanical path and does not refract like light.

Why it mattersHow can one flat water surface make a fish look shallower, fool a spear, squeeze the sky and distort photos?

Start with the essentials

Focus question: How can one flat water surface make a fish look shallower, fool a spear, squeeze the sky and distort photos?
One-sentence intuition: The familiar rule that water looks three quarters as deep holds only when you look straight down. 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 sideways offset of the point where the ray leaves the surface, so the drawn envelope of neighboring rays moves smoothly as that point moves.

Core mathematical model

Apparent image off axis, in the meridional plane

\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 emerging rays in the meridional plane. Looking straight down, it equals the real depth divided by the relative index; toward the critical angle, the image approaches the surface. Off axis, a flat refracting surface does not form a perfect point image.

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 whole hemisphere above the water is squeezed inside this half angle. Outside the circle, an underwater observer sees light from below that has been totally internally reflected, not transmitted sky light.

Shooting equation for a parallel flat port

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 sets the angles in water, glass and air. Solving for the ray that reaches the camera shows why a straight pinhole ray crosses a flat port at the wrong place.

Common difficulties

Using one shallow virtual fish for every viewing angle

Typical misconceptionAn underwater object always looks exactly three quarters as deep as it is, so an off-axis ray can be drawn toward that fixed point.

Better mental modelThat ratio holds only near the axis. The exact image formed by neighboring rays depends on the viewing angle, and in three dimensions it splits into principal images. This Lab draws only the meridional envelope and labels it as such.

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. Compare the real fish, the violet envelope, the paraxial reference and the exact depth readout.
What to observe: Near normal view, the exact and paraxial depths almost agree. At larger angles the exact meridional image rises quickly, and the backward extensions of neighboring rays no longer meet at one perfect point.
02
Scene 2: Aim below the virtual fish
In the aiming scene, keep the directions straight: light travels from the fish to the eye, while the hypothetical spear travels from the launcher into the water. Compare the naive impact with the gold correction.
What to observe: A projectile aimed straight at the virtual fish crosses the surface without bending and lands beyond the real target. You have to aim below the image, and the correction grows with the sight offset and the depth.
03
Scene 3: Snell window and flat port
Orbit below the Snell cone and move the selected sky ray, then look at the flat-port cutaway at the side. Pick out each part in turn: the gold horizon rim, the magenta total reflection, the green camera rays that bend twice, and the violet pinhole error.
What to observe: The horizon maps onto the critical rim. Parallel glass keeps the tangential momentum the same through both port faces, yet the resulting rays do not follow a straight-line pinhole model with a single viewpoint.