Simple Circular Curve Geometry – Mid-ordinate (Distance from Long Chord to Curve Mid-point) For a simple circular curve of radius R and central (deflection) angle Δ, what is the distance from the mid-point of the curve to the long chord (i.e., the mid-ordinate)?

Introduction / Context:
Highway and railway curve setting frequently requires quick geometric quantities such as long chord, mid-ordinate, and external distance. The mid-ordinate is the maximum offset between the circular curve and its long chord, occurring at the curve’s mid-point. Knowing its closed-form expression speeds up field checks and design verification.

Given Data / Assumptions:

• Simple circular curve with radius R.
• Central (deflection) angle of the curve = Δ (in degrees or radians consistently).
• Long chord is the chord joining the tangent points; its length is 2 * R * sin(Δ / 2).

Concept / Approach:

In the isosceles triangle formed by the two radii to the tangent points and the long chord, the line from the circle center to the chord mid-point is perpendicular to the long chord. The mid-ordinate m is the difference between the radius and the adjacent segment along the radius projected to the chord mid-point, which introduces the cosine of Δ/2.

Step-by-Step Solution:

Verification / Alternative check:

For small Δ, use cos(Δ/2) ≈ 1 − (Δ^2 / 8) (in radians), giving m ≈ R * (Δ^2 / 8). This matches the small-angle behavior used for rapid estimates.

Why Other Options Are Wrong:

(b) and (e) use Δ instead of Δ/2, which overestimates; (c) gives half-chord length, not mid-ordinate; (d) is the external distance approximation for some layouts, not the mid-ordinate formula.

Common Pitfalls:

Mixing Δ with Δ/2; confusing mid-ordinate with external distance; forgetting units (degrees vs radians) when using approximations.

Final Answer:

R * (1 - cos(Δ / 2))