Lemniscate transition used throughout to connect two parallel roads: What is the maximum polar angle of the curve?

Introduction / Context:
Lemniscate curves (Bernoulli lemniscate) are used as transition curves in special cases, particularly for road/rail connections needing gentle curvature variation and symmetry, such as links between parallel alignments.

Given Data / Assumptions:

• Curve type: Lemniscate (used as a transition throughout).
• Objective: identify its limiting polar angle when used to bridge parallel roads.

Concept / Approach:
The lemniscate has a characteristic polar equation and symmetry that makes the maximum polar angle for a transition between parallels equal to 45 degrees. This reflects the locus where curvature properties satisfy the “equal angle” requirement relative to the axes, enabling smooth reverse curvature behavior suitable for parallel links.

Step-by-Step Solution:
Recognize that for a lemniscate used as a continuous transition, the geometry limits the polar angle.Recall the standard property: θ_max = 45° (π/4 radians).Choose the option with 45°.

Verification / Alternative check:
Geometric construction notes for lemniscates cite θ_max = 45° for symmetrical transitions between parallels to keep curvature variation within comfort limits.

Why Other Options Are Wrong:

• Angles less than 45° do not reach the full symmetrical spread required.
• Angles greater than 45° exceed the standard lemniscate bound for this application.

Common Pitfalls:

• Confusing clothoids (clothoid angle not capped like 45°) with lemniscates.
• Assuming the maximum polar angle is dependent on speed; it is a geometric property here.

Final Answer:
45°