Terminator
What is the terminator?
The terminator is the line between day and night — the circle on Earth where the Sun sits on the horizon. Sunrise is where you cross it going into the light; sunset is where you cross it going out.
A circle, drawn as a wave
On a globe the terminator is a circle: everywhere the Sun is exactly on the horizon. Half the planet is in front of it, half behind. Because sunlight arrives almost in parallel, that split is always half-and-half, whatever the date.
The map at the top of this site is flat, so the circle is stretched into a curve. Near an equinox the curve runs pole to pole. Near a solstice it leans, and one polar cap stays in daylight while the other stays dark.
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How fast it moves
The line itself is not a thing racing over the ground. Earth is turning under a Sun that, for this purpose, sits still. At the equator that turn is about 1,670 km/h — 15 degrees of longitude per hour, which is why time zones are an hour wide.
This map solves the terminator per meridian: for each longitude, where is the Sun at altitude 0°? That is one equation in latitude. The shaded night is those latitudes closed toward the dark pole. Twilight is the same solve at −18°.