You are standing in the middle of a straight path of width W (the edges of which are parallel), looking along the midline. (It’s a lot less dangerous than standing in the middle of train tracks, but the principle still applies there as well.) The horizontal distance along the path to any point on the midline is y. Perspective, of course, makes the edges (and shadows here) appear to converge to a point (the apex angle, A).

How does A depend on W, and the height h of your eyes above the ground, as your “angle of dip” (θ) approaches zero (i.e., y approaches infinity)? Under these circumstances, A(θ) is defined as the derivative /, where ω(θ) is the angle subtended at your eye by the path width W.

Specifically, show using elementary geometry that ω...

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