Geometry of parallel sidings and crossings: If D is the distance between two parallel sidings and β is the limiting crossing angle, what is the distance between the noses of the two crossings measured parallel to the main track?

Introduction / Context:
Turnout and crossing geometry often requires projecting perpendicular offsets along a track line. When two sidings are parallel and separated by a distance D, designers need the longitudinal spacing between crossing noses.

Given Data / Assumptions:

• Sidings are parallel, separated by D (perpendicular to main track).
• Crossing angle with main is β (acute angle).
• Required: component along main track.

Concept / Approach:
Basic trigonometry: if a perpendicular gap D is projected along a line inclined at angle β, the along-track distance equals the perpendicular distance divided by tan β, i.e., D / tan β, which is D cot β.

Step-by-Step Solution:
Let longitudinal distance along main = L.tan β = opposite / adjacent = D / L.Therefore, L = D / tan β = D cot β.

Verification / Alternative check:
Check limiting behavior: for smaller β (flatter), tan β decreases and L increases, which is physically consistent.

Why Other Options Are Wrong:

• D sin β, D cos β, D sec β: incorrect projections for the required along-track component.
• D tan β: gives the transverse, not longitudinal distance.

Common Pitfalls:
Swapping tan and cot; mixing up perpendicular and parallel components.

Final Answer:
D cot β