The celestial equator is a great circle on the imaginary celestial sphere, in the same plane as the Earth's equator. In other words, it is a projection of the terrestrial equator out into space.1 As a result of the Earth's axial tilt, the celestial equator is inclined by 23.4° with respect to the ecliptic plane.
An observer standing on the Earth's equator visualizes the celestial equator as a semicircle passing directly overhead through the zenith. As the observer moves north (or south), the celestial equator tilts towards the opposite horizon. The celestial equator is defined to be infinitely distant (since it is on the celestial sphere); thus the observer always sees the ends of the semicircle disappear over the horizon exactly due east and due west, regardless of the observer's position on Earth. (At the poles, though, the celestial equator would be parallel to the horizon.) At all latitudes the celestial equator appears perfectly straight because the observer is only finitely far from the plane of the celestial equator but infinitely far from the celestial equator itself.2
Celestial bodies other than Earth also have similarly defined celestial equators.
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