Horizontal plane in a rotating system.
We are revolving with the earth, and can't differentiate between the attractive force of the earth and the centrifugal force that accompanies revolution, and only feel the net effect of those two forces as gravity. In other words, for us, "straight down" does not point to the center of the earth. Further, a horizontal plane is a plane perpendicular to the combined attractive and centrifugal forces of the earth, so the earth, for example, seems like an ellipsoid.
On a rotating table, a horizontal plane is perpendicular to a combination of centrifugal force and gravity, becoming a paraboloid. When you swing a pendulum on a rotating table, the plane of oscillation is spherical, however, when the period of the pendulum and the period of the rotating table coincide, the plane of oscillation becomes equivalent to a horizontal plane. (At the center of rotation, the curvature becomes equal.) In other words, the weight of the pendulum is always on a horizontal plane, and always pointing "straight down".
