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The formula for external length can be given as __________
E=2R×tan2∆
E=2R×sin2∆
E=R(sec2∆−1)
E=R(1−cos2∆)
E=R(sec2∆−1)
In highway and railway curve surveying, the external distance (or external secant length) E is the distance from the point of intersection (PI) to the apex of the simple circular curve. It represents the maximum distance from the tangents to the curve measured along the bisector of the deflection angle Δ. The correct expression is derived geometrically as E=R(sec2Δ−1).
In highway and railway curve surveying, the external distance (or external secant length) E is the distance from the point of intersection (PI) to the apex of the simple circular curve. It represents the maximum distance from the tangents to the curve measured along the bisector of the deflection angle Δ. The correct expression is derived geometrically as E=R(sec2Δ−1).
E=R(sec2Δ−1) — External distance (Apex distance)
M=R(1−cos2Δ) — Mid-ordinate (Versed sine)
T=Rtan2Δ — Tangent length
L = \frac{\pi R \Delta}{180° — Length of curve
In a simple circular curve of radius R and deflection angle Δ, the distance from the center of the curve O to the Point of Intersection V is OV=Rsec2Δ. Since the distance from O to the curve apex C is the radius R, the external length E=VC=OV−OC=Rsec2Δ−R=R(sec2Δ−1).
External distance E is the distance from the Point of Intersection (PI) to the midpoint/apex of the curve.
It is also called the secant distance because it is derived using the secant trigonometric ratio in the right-angled triangle formed by the center, tangent point, and PI.
As deflection angle Δ increases, external distance E increases exponentially.
Determining clear spacing requirements between the intersection point and the highway/railway centerline.
Designing circular horizontal curves to ensure proper clearance around obstacles at hill crests or intersections.
Option A (2Rtan2Δ) is incorrect; the tangent length is T=Rtan2Δ.
Option B (2Rsin2Δ) is incorrect; the long chord length is L=2Rsin2Δ.
Option D (R(1−cos2Δ)) represents the Mid-ordinate (M), not the external distance.
C is correct — The external distance E of a simple circular curve is given by E=R(sec2Δ−1).
Remember the relation between External Distance E and Mid-Ordinate M: E=R(sec2Δ−1) while M=R(1−cos2Δ). Notice how sec is used for E (outside the arc) and cos for M (inside the arc).
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