For a lemniscate transition used in highway geometry, if the maximum polar angle is α and the matched circle radius is R at the junction, the maximum radial distance along the lemniscate is:

Introduction / Context:
Lemniscate curves are sometimes used as transition elements where curvature needs to vary in a prescribed manner. Unlike a simple clothoid, a lemniscate can provide specific curvature properties over larger deflection angles. Understanding the relationship between polar angle and radial distance helps in geometric checks and staking out the curve in the field.

Given Data / Assumptions:

• A lemniscate transition is adopted with maximum polar angle α.
• R is the matched circular-arc radius at the junction with the main circular curve.
• Standard lemniscate proportioning used for highway transition design.

Concept / Approach:
In the polar formulation of a lemniscate transition, radial distance r is a function of the polar angle and the matching radius R. For the selected family used in practice, the maximum radial reach along the lemniscate develops as a sine function of twice the polar angle, scaled by a constant factor tied to R.

Step-by-Step Solution (parameter reasoning):
Relate the transition geometry to the adjoining circle of radius R at the junction.Use the standard lemniscate proportion that produces r proportional to sin(2α) at the limit angle.This leads to r_max = 3 * R * sin(2α) for the adopted family.

Verification / Alternative check:
Design handbooks tabulate lemniscate offsets and polar relations. The dependency on sin(2α) matches the symmetry of the curve lobes and its curvature progression, providing a quick check during layout.

Why Other Options Are Wrong:

• 3 R sin α and 3 R sin(α/2): wrong angular argument; do not reflect the curve’s two-lobed symmetry.
• 3 R sin2α (without space) is dimensionally the same as option (c) but is a typographic variant; the intended correct relation uses sin(2α).
• None of these: incorrect since a standard closed-form exists for the selected family.

Common Pitfalls:

• Confusing lemniscate properties with clothoid (Euler spiral) formulas.
• Mixing degrees and radians in trigonometric evaluation.

Final Answer:
3 R sin 2α