Visualiser series · 01

Total Internal Reflection.

The waveguide pendants on the bench hold their own light. Drop a single warm-white LED into a resin form and most of the light cannot get out — it bounces off the inside of every surface, runs down the length of the piece, and leaks only where the geometry tells it to. The mechanism is total internal reflection, and the load-bearing number is the critical angle: θᴄ = arcsin(n₂ / n₁). For the studio’s standard resin (n ≈ 1.50) looking out into air (n ≈ 1.00), that is 41.81° from the surface normal. Any internal ray hitting the wall at a steeper angle than that has nowhere to go but back into the resin.

This page is the maths made tangible. Drag the incident angle below. Watch the refracted ray dim and vanish the moment the angle crosses the critical line; watch the reflected ray go from a faint companion to the entire story. Swap the materials and the threshold moves — diamond traps at 24.4°, water at 48.6°, ordinary glass somewhere in between. The Fresnel readout gives the honest percentage of light that reflects at any incident angle, not just above the critical line.

The prose companion is Why the Pendant Glows From the Inside — that piece is the long-form derivation of every number on this page. The wavelength side of the same family of mechanisms is in Colour Without Pigment.

Parameters

n₁ — source medium index1.50

n₂ — target medium index1.00

Incident angle θᵢ — from the normal50.0°

Presets

Readout

Critical angle θᴄ: 41.81°
State: TIR engaged — light trapped
Refraction angle θₜ: — (no transmitted ray)
Fresnel reflectance R: 100.0%
Transmittance T: 0.0%

R is the unpolarised average of the s- and p-polarisation Fresnel reflectances. Above θᴄ the value locks at 100% — total internal reflection. Below the critical angle R rises from a few per cent at normal incidence to a brief dip at Brewster’s angle and then climbs steeply toward grazing.

Where this matters in the studio

Every waveguide piece on the bench — every pendant, every cuff, every wall array — rests on the arithmetic above. The critical angle sets the wall angles that trap light, the minimum bend radius for printable channels, and the surface treatments that let the trapped light escape where the design wants it.

— Why the Pendant Glows From the Inside — the long-form derivation, with the minimum-bend-radius formula and the compound-loss arithmetic.
— Colour Without Pigment — the wavelength side. Thin-film interference, diffraction gratings, photonic crystals: same family of tricks, different mechanism.
— The stack — waveguide section — the kit and material list the bench uses to turn this physics into objects.

The Fresnel reflectance shown in the readout is the unpolarised average of the s- and p-polarisation Fresnel coefficients — not Schlick’s approximation, which loses accuracy near Brewster’s angle and at grazing incidence. The maths is in lib/visualiser/tir-math.ts and is pure-function and testable. First entry in a /visualiser/* route family with a consistent shape: page chrome + R3F canvas + controls block + readout. Future entries will follow the same skeleton.