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The formula for mid ordinate length can be given as __________
M=2R×tan2∆
M=2R×sin2∆
M=R(sec2∆−1)
M=R(1−cos2∆)
M=R(1−cos2∆)
The mid-ordinate (M) of a simple circular curve is the ordinate from the midpoint of the long chord to the midpoint of the curve. It is also known as the versed sine of the curve. The correct expression for calculating the mid-ordinate length is M=R(1−cos2Δ).
The mid-ordinate (M) of a simple circular curve is the ordinate from the midpoint of the long chord to the midpoint of the curve. It is also known as the versed sine of the curve. The correct expression for calculating the mid-ordinate length is M=R(1−cos2Δ).
M=R(1−cos2Δ) — Length of Mid-Ordinate
T=Rtan2Δ — Tangent Length
E=R(sec2Δ−1) — Apex Distance (External Distance)
L = R \times \Delta \times \frac{\pi}{180° — Length of Curve
Lc=2Rsin2Δ — Length of Long Chord
In a simple circular curve, the distance from the center of the curve O to the midpoint of the curve is the radius R. The distance from the center O to the midpoint of the long chord is Rcos(2Δ). Subtracting this from the radius gives M=R−Rcos(2Δ)=R(1−cos2Δ).
The mid-ordinate is the perpendicular distance between the midpoint of the long chord and the apex of the curve.
Mid-ordinate M is also called the versed sine of half the deflection angle.
Setting out simple circular curves in field surveying using the offset from long chord method.
Determining sight distance requirements on horizontal curves in highway engineering.
Option A (2Rtan2Δ) is incorrect; tangent length is Rtan2Δ.
Option B (2Rsin2Δ) represents the total length of the long chord (Lc).
Option C (R(sec2Δ−1)) represents the apex distance or external distance (E).
D is correct — The formula for mid-ordinate length is M=R(1−cos2Δ).
Remember that external distance E uses (sec2Δ−1), while mid-ordinate M uses (1−cos2Δ). Since cos and sec are reciprocals, note that E=R(1+Rcos(Δ/2)M−1) relates both parameters geometrically.
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