Local tangent drop
At surface-arc distance d and radius R, tangent drop is R(1 − cos(d/R)). For small d/R, it is approximately d²/(2R). With d in miles, that local approximation is close to eight inches per mile squared. It is not the hidden height behind an observer’s horizon.
An observer adds a tangent triangle
For eye height h, geometric horizon dip is acos(R/(R+h)). The straight tangent distance is √((R+h)² − R²); the surface arc is R times the dip angle in radians. The two distances agree closely at small heights but have distinct definitions.
A chord has two endpoints
The midpoint bulge above a chord joining two surface endpoints is R(1 − cos(d/(2R))). At short distance it is approximately d²/(8R), one quarter of the tangent-drop approximation. Target visibility uses observer and target heights rather than this chord bulge alone.