Circular Curves – Approximate Formula for Perpendicular (Radial) Offsets from the Tangent For setting out a simple circular curve, what is the commonly used approximate expression for the perpendicular (radial) offset y from the tangent at a point located a distance x along the tangent from the point of tangency (radius = R)?

Introduction / Context:
In highway and railway curve setting, quick field layout often relies on approximate formulas for offsets. Perpendicular (sometimes called radial) offsets from the tangent are especially handy near the tangent point, where the curve deviates gradually from the tangent. This question asks you to recall the standard approximation used by surveyors.

Given Data / Assumptions:

• Simple circular curve of radius R.
• Point on the tangent located at distance x from the point of tangency T.
• Offset y is measured perpendicular to the tangent to reach the curve.

Concept / Approach:

From circle geometry, the exact offset from the tangent at distance x is y_exact = R − √(R^2 − x^2). For x ≪ R (near T), a binomial expansion gives √(R^2 − x^2) ≈ R − x^2/(2R). Therefore, y ≈ x^2/(2R). This approximation is robust for small x/R and is widely used for peg setting at short intervals from T in the field.

Step-by-Step Solution:

Verification / Alternative check:

Numerical test: for R = 300 m and x = 10 m, exact y ≈ 0.1667 m; approximation gives x^2/(2R) = 100/600 ≈ 0.1667 m, confirming high accuracy for small x.

Why Other Options Are Wrong:

y = x^2/R and y = x^2/(3R) over/under-estimate the true offset; the cubic term y = x^3/(6R^2) is not the leading-order term; y = 2 R x is dimensionally incorrect and grossly wrong.

Common Pitfalls:

Using the approximation too far from T (large x/R), where errors grow; mixing up perpendicular offsets with radial offsets from the long chord method.

Final Answer:

y = x^2 / (2R)