Turnout geometry (curve lead): For a standard turnout, which of the following is the correct expression for the curve lead (C.L.) in terms of gauge G and crossing angle α (or crossing number N)?

Introduction / Context:
Turnout layout is defined by key geometrical lengths: curve lead (C.L.), switch lead, and overall lead. Correct formulae ensure the turnout fits longitudinally and laterally within available track space.

Given Data / Assumptions:

• G = gauge (centre-to-centre distance between stock and turnout track).
• α = crossing angle; N = crossing number such that N = cot(α/2).
• “I” would denote a layout parameter not universally standard; we focus on widely adopted expressions.

Concept / Approach:
The curve lead is the length along the turnout curve needed to achieve the lateral offset equal to the gauge at the crossing. Using simple geometry with small-angle assumptions, C.L. = G * cot(α/2). Since N = cot(α/2), C.L. = G * N.

Step-by-Step Solution:
Relate crossing number N to angle: N = cot(α/2).Write curve lead as C.L. = G * N = G * cot(α/2).Hence option (a) expresses the correct relation explicitly.

Verification / Alternative check:
Using a typical BG turnout with G ≈ 1.676 m and N = 12, C.L. ≈ 1.676 * 12 ≈ 20.1 m, which matches standard tables.

Why Other Options Are Wrong:

• 2 * I * G and 2 * G * N represent other lengths or are not standard for C.L.; “2” factor overestimates the curve lead.
• All the above: Incorrect; only (a) (or equivalently G * N) is valid.

Common Pitfalls:
Confusing curve lead with overall lead; misusing crossing number definitions; mixing degrees and radians.

Final Answer:
C.L. = G * cot(α/2)