Crossover geometry: For two parallel tracks at center-to-center distance D, gauge G, and crossing angle α, what is the expression for the distance between the theoretical noses of the two crossings measured parallel to the tracks?

Introduction / Context:
Designing a crossover between parallel tracks requires careful geometry of turnouts and crossings. The spacing between theoretical noses of the two crossings, measured parallel to the track centerline, depends on track spacing D, gauge G, and the crossing angle α.

Given Data / Assumptions:

• D = center-to-center spacing of the two main tracks.
• G = track gauge.
• α = angle of crossing for the turnouts.
• Equal crossing angles and symmetric arrangement.

Concept / Approach:
The geometry resolves transverse separations into longitudinal distances using trigonometry. Because the nose-to-nose line is aligned parallel to the tracks, the relevant transverse components project by a factor of cot α. Adjustments for gauge are included due to the nose being at the gauge line, not track center.

Step-by-Step Solution:
1) Represent the transverse offset between noses as the basic track spacing modified by gauge terms at a crossing.2) Determine the effective transverse separation: (D - G + G sec α).3) Project this separation along the parallel (track) direction using cot α to obtain the longitudinal nose spacing.4) Final expression: (D - G + G sec α) cot α.

Verification / Alternative check:
Check limiting cases: As α becomes small (flatter crossing), cot α grows and the nose spacing increases, which is physically consistent for flatter crossings requiring longer leads.

Why Other Options Are Wrong:

• Signs in the gauge terms are incorrect in options (b) and (c), giving wrong effective separation.
• Using tan α in (c) projects in the wrong sense; the dimension should scale with cot α for small angles.
• (d) overestimates by adding both G and G sec α.

Common Pitfalls:

• Ignoring that noses are located at gauge lines, not at track centers, leading to omission of gauge corrections.

Final Answer:
(D - G + G sec α) cot α.