Crossing geometry: setback between theoretical and actual nose For a common crossing, the practical (actual) nose is rounded. The distance between the theoretical nose of crossing and the actual nose is proportional to nose thickness. It is given by which relationship (α = crossing angle)?

Introduction / Context:
In a turnout crossing, the theoretical nose is an ideal sharp point where gauge lines intersect. Practically, the nose is rounded for strength and to avoid wheel impact, so it is set back from the theoretical point. Designers need a simple relation to compute this setback for drawings and quality control.

Given Data / Assumptions:

• Crossing angle is α.
• Nose tip is rounded; thickness at nose is known.
• Setback is measured along the gauge line direction used in standard drawings.

Concept / Approach:
By resolving the geometry of the rounded nose into the crossing angle, the linear setback along the gauge line becomes proportional to the nose thickness multiplied by cot α. This comes from right-triangle relations between the normal to the gauge line, the nose thickness, and the included angle at the theoretical point.

Step-by-Step Solution:
Model the rounded tip as a finite thickness replacing the sharp theoretical apex.Project this thickness along the gauge line at angle α.Using trigonometry, setback = thickness × cot α.

Verification / Alternative check:
Standard turnout drawing formulae and inspection templates use the cotangent relation for setback from theoretical to actual nose, matching workshop practice for machining and fit-up.

Why Other Options Are Wrong:
tan α, sin α, cos α: Do not represent the correct projection geometry for the setback along the gauge line; the cotangent relation is the one used in standard detailing.

Common Pitfalls:

• Measuring setback in the wrong direction or from the wrong gauge face.
• Confusing crossing angle definition (with respect to gauge line) leading to wrong trigonometric function.

Final Answer:
Nose thickness × cot α