Crossing geometry with parallel sidings If D is the perpendicular distance between two parallel sidings and α is the acute crossing angle, what is the distance between the noses of the crossings measured parallel to the main track?

Introduction / Context:Turnout and crossing design requires translating a perpendicular separation into a longitudinal spacing along the main track. Trigonometric relations provide the conversion using the crossing angle.

Given Data / Assumptions:

• D = perpendicular spacing between parallel sidings.
• α = acute angle of crossing with the main track.
• Need: distance between noses along the main track direction.

Concept / Approach:Draw a right triangle where the opposite side is D (perpendicular gap) and the angle at the main track is α. Then tan α = opposite/adjacent = D/L, where L is the required longitudinal spacing. Rearranging gives L = D / tan α = D cot α.

Step-by-Step Solution:Let L = distance along main track.tan α = D / L.Therefore, L = D / tan α = D cot α.Hence the correct expression is D cot α.

Verification / Alternative check:As α becomes smaller (flatter crossing), tan α decreases, so L increases—physically consistent since a flatter crossing requires longer longitudinal development.

Why Other Options Are Wrong:

• D tan α: gives transverse increment for a given longitudinal run, not the required along-track spacing.
• D sec α or D cosec α: do not match the right-triangle projection for this geometry.
• None: invalid because a correct expression exists.

Common Pitfalls:Interchanging tan and cot; mixing up which side of the triangle corresponds to D or L.

Final Answer:D cot α