Which of the following formula for the perpendicular offset can be given as

Question

Examoogle · Accepted Answer

Option B: $O_x = R - \sqrt{R^2 - x^2}$

Explanation & Concepts:
In highway and railway engineering, setting out simple circular curves by offsets from the long chord is a common field method. The perpendicular offset $O_x$ at any distance $x$ measured along the long chord from its midpoint (where the versine or mid-ordinate $O_0$ lies) is calculated as $O_x = \sqrt{R^2 - x^2} - (R - O_0)$, which simplifies to $O_x = R - \sqrt{R^2 - x^2}$ when referenced directly from the tangent arc baseline definition.

Working Principle: Consider a circular curve of radius $R$ centered at $O$. Let the midpoint of the long chord be $M$, where the offset is maximum ($O_0$). For a point at distance $x$ from $M$ along the long chord, the perpendicular offset to the curve is $O_x$. Using the right-angled triangle formed by the radius, the offset, and the distance $x$, the geometry yields $(R - O_x)^2 + x^2 = R^2$, which solves directly to $O_x = R - \sqrt{R^2 - x^2}$.

Key Formulas: $O_x = R - \sqrt{R^2 - x^2}$ — Exact perpendicular offset from tangent; $O_x \approx \frac{x^2}{2R}$ — Approximate formula for perpendicular offset when $x \ll R$; $O_0 = R - \sqrt{R^2 - \left(\frac{L}{2}\right)^2}$ — Versine or mid-ordinate formula

Key Points: The maximum offset occurs at the center of the chord ($x = 0$) and is called the mid-ordinate ($O_0$).; Perpendicular offsets can be taken either from the long chord or from the tangent line.; The exact expression $O_x = R - \sqrt{R^2 - x^2}$ is derived directly from the Pythagorean theorem on the circle radius triangle.

Advantages: Simple geometric setting out method without requiring advanced optical instruments like total stations.; Highly suitable for short curves such as street intersections and circular culverts.

Disadvantages: Accumulation of errors can occur for long curves.; Requires precise right-angle measurements in the field which can be tedious over long distances.

Applications: Setting out simple circular curves in road layout and railway tracks.; Aligning kerb lines and circular boundary structures in civil site construction.

Additional Info: Option A gives a negative value since $\sqrt{R^2-x^2}. Option C corresponds to an incorrect hypotenuse calculation $\sqrt{R^2+x^2}$, which lies outside the circle geometry.. Option D adds the radical term, resulting in values exceeding $R$, which places points far outside the geometric boundary.

Answer: B is correct — The perpendicular offset $O_x$ at distance $x$ from the midpoint/tangent point is geometrically derived as $O_x = R - \sqrt{R^2 - x^2}$.

Exam Tip: Remember for GATE/ESE exams: when $x$ is very small compared to $R$, use the binomial approximation $O_x \approx \frac{x^2}{2R}$. Use the exact formula $R - \sqrt{R^2 - x^2}$ whenever precise numerical calculations are requested.

Examoogle · Answer

Option A: $O_x = \sqrt{R^2 - x^2} - R$

Examoogle · Answer

Option C: $O_x = \sqrt{R^2 + x^2} - R$

Examoogle · Answer

Option D: $O_x = R + \sqrt{R^2 - x^2}$

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