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Examoogle • User • info@examoogle.com • EE-2024-8821
Chapter 1 of 12 • Page 1 of 248🔒 Protected PDF • Watermarked
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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).