Linear Interpolation Between Calibration Points

An optical pyrometer sees a piece of hot steel glowing at 645 nanometres and needs to report a temperature. You calibrated the instrument at known temperatures — a blackbody furnace at 800°C peaked at 620 nm, at 900°C it peaked at 660 nm — but 645 nm wasn’t on your list. You could measure every nanometre from 400 to 2000, but that’s a lot of furnace time and a lot of storage.

Linear interpolation assumes the relationship between your calibration points is a straight line. It’s wrong — blackbody radiation follows Planck’s law, which is decidedly curved — but if your points are close enough, the error is smaller than your sensor noise. For a pyrometer reading every 20 nm across its range, you’d store fifty wavelength-temperature pairs instead of sixteen hundred, and the lookup stays fast.

Here’s the idea in Python, given a calibration table sorted by wavelength:

def interpolate_temp(calibration, wavelength):
    for i in range(len(calibration) - 1):
        wl0, temp0 = calibration[i]
        wl1, temp1 = calibration[i + 1]
        if wl0 <= wavelength <= wl1:
            t = (wavelength - wl0) / (wl1 - wl0)
            return temp0 + t * (temp1 - temp0)
    return None

cal = [(620, 800), (660, 900), (705, 1000)]
print(f"{interpolate_temp(cal, 645):.1f}°C")  # 862.5°C

Zig makes the types explicit and the performance predictable — useful when this runs in a 1978 microcontroller loop at 10 kHz:

fn interpolate(cal: [][2]f32, wl: f32) ?f32 {
    for (cal[0..cal.len-1], 0..) |pt, i| {
        const next = cal[i + 1];
        if (pt[0] <= wl and wl <= next[0]) {
            const t = (wl - pt[0]) / (next[0] - pt[0]);
            return pt[1] + t * (next[1] - pt[1]);
        }
    }
    return null;
}

The assumption that the curve is locally linear breaks down at the edges of your range or across phase transitions, but for routine measurement of glowing metal, it’s been good enough for fifty years. Most modern pyrometers still do this, just with tighter point spacing and better sensors.