Distance to the horizon point on a tilted photograph On a tilted photograph with tilt angle θ, what is the distance from the principal point to the horizon point along the principal line?

Introduction / Context:Tilt introduces a visible horizon on an aerial photograph. The location of the horizon along the principal line is a key construction needed for tilt corrections and for understanding scale variations across the image.

Given Data / Assumptions:

• Photograph of focal length f with tilt angle θ about an axis in the plate.
• Definitions: principal point is the perpendicular foot of the perspective center on the image plane; principal line is the line through principal and nadir/plumb points.
• Level terrain so the horizon is the image of the plane at infinity.

Concept / Approach:For a tilted photograph, the image of the horizon lies on the principal line at a distance from the principal point given by f cot θ, measured in the direction of tilt. This follows from the perspective mapping of a horizontal plane under a rotation by θ.

Step-by-Step Solution:Model the tilt as a rotation of the camera by θ.The horizon corresponds to rays parallel to the ground plane; their intercept on the image plane is at distance f cot θ.Therefore, distance PP′ (principal point to horizon point) = f cot θ along the principal line.

Verification / Alternative check:Limiting cases: as θ → 0 (nearly vertical), cot θ → ∞, so the horizon recedes to infinity (not visible), which matches expectations. As θ increases, the horizon moves closer toward the principal point.

Why Other Options Are Wrong:

• f tan θ, f sin θ, f cos θ, and f sec θ do not satisfy the correct limiting behavior and geometry of the horizon placement for small θ.

Common Pitfalls:Confusing the horizon point with the isocentre or plumb point; only the horizon location follows the f cot θ relation on the principal line.

Final Answer:f cot θ