The Knife-Edge Is a Derivative

Point a razor blade at the focus of a mirror and you’ve built a differentiator. That’s the part of schlieren imaging I was slow to feel: the knife-edge doesn’t record the air, it records the rate of change of the air’s refractive index. Where density is flat, every ray lands where it always lands and the frame sits at half-brightness. Where a plume bends light — a gradient — some rays clear the blade and some get eaten, so that patch goes bright or dark in proportion to the slope. Rotate the blade ninety degrees and you sample the slope along the other axis instead.

That is a convolution kernel. The oldest one in the book, in fact: the central difference [-1, 0, 1], dragged across a row of pixels. This is the arithmetic that machine-vision rigs were burning into hardware all through the 1980s to find edges in a frame. Optics does it continuously and at the speed of light; code does it one little triple at a time.

row = [10, 10, 12, 40, 80, 82, 30, 10, 10]

row
|> Enum.chunk_every(3, 1, :discard)
|> Enum.map(fn [a, _b, c] -> c - a end)
|> IO.inspect(label: "gradient")
# gradient: [2, 30, 68, 42, -50, -72, -20]

The chunk_every(3, 1) slides a three-wide window down the row, and c - a is the kernel. Feed it a smooth ramp and you get a flat non-zero number; feed it an edge and you get a spike exactly where the brightness jumps — the same spike the razor turns into a bright rim on a candle plume. The sign even tells you which way density is changing, which is the bright side versus the dark side of the blade.

Bash does the identical arithmetic with array indices, and I find that oddly pleasing, because it makes the window completely literal: i-1 and i+1 are the two rays straddling the edge.

#!/usr/bin/env bash
row=(10 10 12 40 80 82 30 10 10)
for ((i = 1; i < ${#row[@]} - 1; i++)); do
  printf '%d ' $(( row[i+1] - row[i-1] ))
done
echo
# 2 30 68 42 -50 -72 -20

Same numbers, both of them. The big positives are a rising edge, the big negatives a falling one — a one-dimensional slice of exactly what the mirror threw on my wall for four seconds yesterday before I fumbled the photograph. I still haven’t caught that plume on a sensor. But I can print its derivative all day.