Caption: Illustrated are 3 types of horizon: true horizon, visible horizon, and celestial horizon (AKA astronomical)---which we usually just call horizon for short.
Features:
The celestial horizon is the limit of the true horizon when the observing point is actually on the sphere for which it is specified.
The celestial horizon is usually what we always mean in astronomy---except when we don't.
Let the radius of a exactly spherical astronomical object be R, the height of the observation point be h, and the line-of-sight distance to the true horizon be d.
The angle at the horizon between a radius and the line of sight is a right angle since the line of sight is tangent to the spherical surface of the astronomical object at the true horizon by definition.
From the Pythagorean theorem, we find
Using the approximate formula we find the following reference results given in Table: True Horizon Reference Distances below for a fiducial tall human of observing from h = 2 meters.
Table: True Horizon Reference Distances
Astronomical Mean Radius True Horizon Distance True Horizon Distant Object Zeroth Order Formula Exact Formula with h = 2 m with h = 2 m (km) (km) (km)
General R 5.0482*sqrt(R/R_⊕) 5.0482*sqrt(R/R_⊕)*sqrt(1+(0.002 km)/(2R)) Earth 6371.0 5.0482 5.0482 Mars 3389.5 3.6821 3.6281 Moon 1737.1 2.6360 2.6360 Ceres 473 1.38 1.38 1950 DA 1.1 0.066332 0.066363
In fact, it seems almost impossible to notice the curvature of the Earth just from looking at the visible horizon even over calm bodies of water where the visible horizon approximates the true horizon.
However, a person with good eye sight under clear, calm weather conditions can see ships, etc., rise or sink below the visible horizon over bodies of water. This sinking-below-the-horizon effect was recorded first by Strabo (63? BCE--24? CE) in Classical antiquity (see Wikipedia: Spherical Earth: Roman Empire). He thought effect had been well know for centuries by sailors and other well informed people. It has probably been known well into prehistory by many people who never bothered to interpret it as implying a spherical Earth.
In the many years of looking over Lake Erie, yours truly---and yours truly was born right on the shoreline---staring at boats (and that country yours truly used to refer to when very young as the United Steaks) yours truly has completely failed to notice the sinking-below-the-horizon effect.
Looking at images taken on the lunar surface, it seems as if the visible horizon is noticeably closer than what we see on Earth (see Wikimedia commons: Category: Astronaut photography on lunar surface).
This may be partially due to the shorter distance to the true horizon, but it may also be partially due to the sharpness of images because of lack of an atmosphere and the stark black of the lunar sky. See the discussion at Wikipedia: Examination of Apollo Moon photographs: Apparently identical backgrounds in Apollo 15 photographs taken at different locations, but it doesn't seem definitive to me.