Lemniscate Curve Geometry – Characteristic Angle Ratio at the Polar Ray End For a lemniscate transition curve, the ratio of (i) the angle between the tangent at the end of the polar ray and the straight, to (ii) the angle between the polar ray and the straight, equals which of the following?

Introduction:
Lemniscate transition curves (e.g., Bernoulli lemniscate used in some route designs) possess useful geometric properties governing how the tangent direction evolves from the initial straight. Recognizing these relationships aids in quick checks of layout angles without re-deriving full equations.

Given Data / Assumptions:

• A lemniscate is used as the transition between a tangent and a circular arc.
• The polar ray is drawn from the point of commencement along a fixed direction (the straight).
• Angles considered: (i) between tangent at the polar-ray end and the straight; (ii) between the polar ray and the straight.

Concept / Approach:

Classical results for the lemniscate show that, at the end of the polar ray, the tangent deflection from the straight equals twice the angle that the polar ray itself makes with the straight. This 2:1 ratio is a handy rule of thumb used in approximate setting-out where lemniscates are adopted instead of clothoids.

Step-by-Step Solution:

Verification / Alternative check:

Standard geometric derivations of the lemniscate in polar coordinates confirm this proportionality at the specified point, which is routinely quoted in transportation engineering references.

Why Other Options Are Wrong:

Other integers or fractional values do not match the inherent geometry of the lemniscate; choosing them would contradict established properties of the curve.

Common Pitfalls:

Confusing lemniscate properties with those of a clothoid (Cornu spiral), which follow different angle–arc length relations.

Final Answer:

2