Geometry of parallel sidings: If D is the distance between two parallel sidings and α is the angle of crossing, what is the distance measured along the gathering line between the noses of crossing?

Introduction / Context:
In yard and siding layout, crossing geometry determines the longitudinal spacing required to transition between parallel tracks. The distance between noses of crossing along the gathering line directly affects land take and layout length.

Given Data / Assumptions:

• Two parallel sidings separated by distance D.
• Angle of crossing with respect to the track axis is α.
• Gathering (lead) line is aligned at angle α to the parallel tracks.

Concept / Approach:
Resolving the perpendicular separation D along a line inclined at angle α, the required length along the gathering line equals the component of the perpendicular distance when projected along the inclined line. Trigonometry gives length = D / tan α = D cot α.

Step-by-Step Solution:
1) Represent the two parallel tracks with perpendicular separation D.2) The gathering line makes angle α with the tracks.3) Distance along gathering line to span D: L = D / tan α = D cot α.

Verification / Alternative check:
Check limiting cases: for small α (shallow crossing), tan α is small and cot α large, hence L becomes large—consistent with practice where flatter crossings need longer leads.

Why Other Options Are Wrong:

• D tan α or D sin α or D cos α give incorrect projections for this geometry.
• D / tan α is algebraically the same as D cot α; the accepted concise form is D cot α.

Common Pitfalls:

• Confusing α as the turnout angle at the switch instead of the crossing angle; in layout calculations, be consistent with defined angles.

Final Answer:
D cot α.