Point a camera at the Moon. This measures what fraction of its disc is lit, by fitting two overlapping circles to the shape — and tells you how much to trust the answer.
The demo renders a Moon of known phase, so you can see the measured number next to the truth.
Added in quadrature: —
Calibrated as a one-sigma band: against test Moons of known phase the truth landed inside it 93% of the time, with the worst case at twice it.
The lit part of the Moon is bounded by two curves: the limb, which is an arc of the Moon’s own circular outline, and the terminator, the line between day and night. The cusps — the two sharp points — always sit at opposite ends of a diameter, so the lit limb is always exactly a half-circle, at every phase.
It is tempting to say the shape is the intersection of two overlapping circles. It cannot be. The intersection of two convex shapes is convex, and a crescent is not convex — it has a dent in it. So the same two circles have to be combined two different ways:
That is the model drawn on the image: a gold circle for the limb, a pink one for the terminator’s circle. Near quarter phase the pink circle is enormous, so past a point the app draws the straight line it is heading towards.
The real terminator is not a circular arc. It is the Moon’s day/night boundary — a circle in 3D — seen at an angle, so it projects to half an ellipse with semi-axes R and R cos i. That gives the textbook illuminated fraction k = (1 + cos i) / 2.
Substituting a circular arc through the same three points changes the enclosed area. The error is a bias, not noise — it always pushes the answer towards 50%:
| True lit fraction | 5% | 20% | 35% | 50% | 65% | 80% | 95% |
|---|---|---|---|---|---|---|---|
| Two-circle model | 6.2% | 22.8% | 37.0% | 50.0% | 63.0% | 77.2% | 93.8% |
| Bias | +1.2 | +2.8 | +2.0 | 0.0 | −2.0 | −2.8 | −1.2 |
It peaks at 2.8 percentage points near 22% and 78% lit, and vanishes exactly at quarter, full and new. Because it is known in closed form, the app does not have to live with it: the headline number comes from fitting the exact half-ellipse, while the circles you see come from fitting the circular model. Both are reported, and the green line on the image is the true terminator — the gap between it and the edge of the blue region is the bias, drawn to scale.
A single accuracy figure would be misleading, because the error sources behave completely differently depending on the phase and on how the photo was taken. The app reports each separately and adds them in quadrature:
The band is calibrated as a true one-sigma, not a guarantee. Against synthetic Moons of known phase — clean, blurred, noisy, small, overexposed and combinations of those — the truth lands inside the band in 114 of 123 cases (93%), with a median error of 0.17σ and a worst case right at 2σ. It has one real-world calibration point too: a hand-held photo taken five and a half hours before the August 2026 full moon, at a true illumination of 99.94%, measured 98.45% — an error of 1.5 points against a quoted ±2.6, or 0.6σ. A further nine synthetic images were declined outright rather than measured, which is the other half of being honest: a crescent thinner than the blur hiding it still leaves a bright blob for the threshold to find, and fitting that blob produced a confident 99% from an image with no Moon left in it. If the brightest region is not clearly brighter than its surroundings, the app says so instead of quoting a number. On a clean image the measurement lands within about half a percentage point of truth.
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