Simple circular curve geometry in highway/rail alignment Let Δ be the total deflection (intersection) angle of a simple circular curve, and T1 and T2 be its tangent points. What is the angle between the initial tangent at T1 and the long chord T1–T2?

Introduction / Context:
In setting out a simple circular curve, various geometric relations between tangents, chords, and central angles are repeatedly used for calculations and field methods (deflection-chord method, long-chord checks). One useful relation is the angle between the initial tangent and the long chord connecting the two tangent points.

Given Data / Assumptions:

• Simple circular curve with intersection (deflection) angle Δ.
• T1 and T2 are tangent points at the beginning and end of the curve.
• Long chord is the straight line T1–T2.

Concept / Approach:

The central angle subtended by the curve equals Δ. The angle that the long chord makes with the tangent at T1 equals half the central angle because the tangent at T1 is perpendicular to the radius at T1, and the chord T1–T2 subtends Δ at the center. By inscribed-angle theorems and symmetry of the circle, the acute angle between the tangent at T1 and chord T1–T2 equals Δ/2.

Step-by-Step Solution:

Verification / Alternative check:

For a small deflection Δ = 0°, tangent aligns with chord → angle 0° (Δ/2 holds). For Δ = 180° (semicircle), chord is a diameter; angle to tangent is 90°, which equals Δ/2 = 90° — consistent.

Why Other Options Are Wrong:

Δ or 2Δ exaggerate the relation; 90° ± Δ/2 are not general expressions for this specific angle.

Common Pitfalls:

Confusing deflection angles used in chord-deflection methods with the geometric angle at T1 to the long chord; mixing interior angle at PI with Δ.

Final Answer:

Δ / 2