The terminator is the boundary between the sunlit and dark halves of the earth. On a globe it is a great circle. On a flat map it becomes a distinctive S-shaped curve, and the shape carries real information.

Step one: find the subsolar point

There is exactly one place on earth where the sun is directly overhead at any given instant. Everything else follows from it.

Latitude of that point is the solar declination — how far north or south of the equator the sun sits today. It swings between about +23.4° in June and −23.4° in December, because the earth's axis is tilted by that amount and points in a fixed direction as the planet orbits.

Longitude is where local apparent noon currently is. That is 180° minus 15° per hour of elapsed UTC, corrected by the equation of time.

Step two: the equation of time

Clock time and sun time are not the same thing, and the gap is not constant.

Two effects cause it. The earth's orbit is an ellipse, so it moves faster when closer to the sun and slower when further away. And the earth's axis is tilted, so the sun's apparent motion is not parallel to the equator. Together these push true solar noon up to about 16 minutes either side of clock noon over the course of a year.

Plot the correction against the date and you get the analemma — the figure-eight you sometimes see printed on globes. Ignore it and your terminator is up to four degrees of longitude out, which is visible on a world map.

Step three: the terminator itself

Once you have the subsolar point at latitude δ and longitude λ₀, the terminator is every point where the sun sits exactly on the horizon. For a longitude λ, the terminator latitude is:

lat = atan( −cos(λ − λ₀) / tan(δ) )

That is the whole thing. Feed it every longitude from −180° to +180°, and you have the curve.

Two consequences fall straight out of the formula:

Near the equinoxes, tan(δ) approaches zero and the expression blows up — the terminator degenerates into a pair of straight meridians running pole to pole. Any implementation has to clamp δ away from exactly zero or the line disappears.

At the solstices, tan(δ) is at its largest, the curve flattens dramatically, and it wraps around one of the poles. That is polar day at one end and polar night at the other.

Why it looks like it moves faster at the equinox

It does not move faster — the earth's rotation is what it is. But near the equinox the terminator is closest to vertical, so as it sweeps west it crosses a given band of latitude in the shortest possible time. Sunrise and sunset happen quickly and change date-to-date rapidly.

Near the solstices the line is at its most slanted, sunrise times drift only a minute or two a day, and high latitudes barely see it move at all.

Sunrise is not when the sun's centre crosses the horizon

One refinement worth knowing, because it is where naive implementations differ from published tables.

The atmosphere refracts light, bending it around the curve of the earth, so the sun is visible before it geometrically rises and after it geometrically sets. The standard convention puts sunrise and sunset at a solar altitude of −0.833°, which accounts for both average refraction and the sun's apparent radius.

Use 0° instead and your sunrise times are a few minutes late everywhere, and significantly off at high latitudes where the sun approaches the horizon at a shallow angle.

Accuracy in practice

The model above — mean longitude, mean anomaly, ecliptic longitude, obliquity — is accurate to well under a degree, which is far below what any world map can render. Full precision ephemerides exist and matter for eclipse prediction; for drawing a day/night line they are pointless.

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Figures on this page are computed from the IANA time zone database. Last generated 2026-08-18.

The day/night terminator, explained