Circular-curve geometry: For a simple circular curve of radius R with deflection (intersection) angle Δ, what is the ratio of the long chord length to the tangent length? Give your answer as a function of Δ.

Introduction / Context:
Highway and railway curve computations repeatedly use relations among the long chord, tangent length, and central angle. Recognizing compact ratios simplifies checking field calculations and designing peg intervals for setting out curves with deflection methods.

Given Data / Assumptions:

• Simple circular curve of radius R.
• Total central (deflection) angle is Δ.
• Long chord T1–T2 spans the curve; tangents from the point of intersection are of equal length.

Concept / Approach:

Standard formulas for a simple circular curve are: long chord = 2 * R * sin(Δ/2) and tangent length = R * tan(Δ/2). The desired ratio is therefore (2 R sin(Δ/2)) / (R tan(Δ/2)) = 2 * sin(Δ/2) / tan(Δ/2) = 2 * cos(Δ/2). This tidy identity is useful for quick checks without substituting R explicitly.

Step-by-Step Solution:

Verification / Alternative check:

Check limiting cases: Δ → 0 ⇒ cos(Δ/2) → 1, so Lc/T → 2, consistent with near-straight geometry where two tangents almost coincide. For Δ = 180°, cos(90°) = 0, ratio vanishes because the tangent length tends to infinity while long chord is finite.

Why Other Options Are Wrong:

sin(Δ/2), cos(Δ/2), tan(Δ/2), and 2 sin(Δ/2) do not match the derived ratio; only 2 cos(Δ/2) is correct.

Common Pitfalls:

Mixing the long chord with a sub-chord; confusing tangent length with external distance or length of curve.

Final Answer:

2 cos(Δ/2)