Crossover geometry with an intermediate straight: If D is the centre-to-centre distance between two parallel tracks of gauge G and the crossing is of number N, what is the total length of the crossover (from commencement to termination)?

Introduction / Context:
Designing a crossover between parallel tracks requires estimating the overall length from the first turnout to the second, including the intermediate straight. The relation depends on the track spacing D, the gauge G, and the crossing number N (where tan α ≈ 1/N for small angles).

Given Data / Assumptions:

• Two parallel tracks, centre spacing = D.
• Track gauge = G.
• Crossing of number N with an intermediate straight between turnouts.

Concept / Approach:
The total crossover length combines: (i) the geometrical lead components associated with each turnout, and (ii) the intermediate straight portion needed for gauge and clearances. In standard derivations using the relationship of offsets and the crossing angle α (with N = 1/tan α), the algebraic sum yields a compact expression in D, G, and N for planning layout length.

Step-by-Step Solution:

Verification / Alternative check:
Dimensional sense-check: each term results in a length; the expression increases with larger N (shallower crossings) and with greater spacing D, aligning with design intuition.

Why Other Options Are Wrong:

• Forms with fewer N-dependent terms underestimate the turnout leads.
• Excess terms or incorrect coefficients do not match standard geometric derivations for crossovers with a straight between turnouts.

Common Pitfalls:

• Confusing geometric turnout lead with switch length or ignoring the intermediate straight portion.
• Misinterpreting crossing number N (it is not the same as cant or versine parameters).

Final Answer:
DN + G (2N + 1 + N^2)