Simple curve geometry for track versine: If L is the chord length of a rail on a circular curve of radius R, what is the versine (mid-ordinate) h?

Introduction / Context:
Versine (sagitta) measurements are widely used in railway track geometry to quantify curvature using a chord and a mid-ordinate. This relation allows quick field checks without directly measuring radius R.

Given Data / Assumptions:

• Circular curve of constant radius R.
• Chord length along the rail = L.
• Versine h is the mid-ordinate from the chord to the arc.

Concept / Approach:
From circle geometry, the exact sagitta is h = R − sqrt(R^2 − (L/2)^2). For railway practice (L small relative to R), the binomial approximation gives the standard working formula h ≈ L^2 / (8R). This is the textbook relation used for versine-based curvature checks.

Step-by-Step Solution:

Verification / Alternative check:
For a sample case (e.g., L = 10 m, R = 1000 m), exact and approximate h values are nearly identical, validating the approximation for typical track work where R is large compared to L.

Why Other Options Are Wrong:

• Forms like 2R/L^2, R^2/(8L), 8R/L are dimensionally inconsistent for length.
• L/(2R) is linear in L; sagitta scales with L^2 for small chords.

Common Pitfalls:

• Using the exact expression unnecessarily in routine checks, adding complexity.
• Confusing versine with mid-chord offset for alignment deviations (a different application).

Final Answer:
h = L^2 / (8R)