Explanation:

To understand why the number of crossings 'N' according to the center line method is given by 1/2 cot(α/2), we need to understand the concept of the center line method and how it relates to the angle of crossing (α).

Center Line Method:
The center line method is a technique used to calculate the number of crossings in a given shape or pattern. It involves drawing a center line through the shape or pattern and counting the number of times the shape or pattern crosses this center line.

Angle of Crossing:
The angle of crossing (α) is the angle between the center line and the line of the shape or pattern at the point of crossing. It represents the direction in which the shape or pattern is crossing the center line.

Now, let's break down the given options and understand why option 'A' is the correct answer.

a) 1/2 cot(α/2):
According to option 'A', the number of crossings 'N' is given by 1/2 cot(α/2).

b) cot(α/2):
Option 'B' suggests that the number of crossings 'N' is given by cot(α/2).

c) cot(α):
Option 'C' suggests that the number of crossings 'N' is given by cot(α).

d) 1/2 cosec(α/2):
Option 'D' suggests that the number of crossings 'N' is given by 1/2 cosec(α/2).

Explanation of the Correct Answer:
The correct answer is option 'A' because it accurately represents the number of crossings 'N' according to the center line method.

In the center line method, the angle of crossing (α) is the angle between the center line and the line of the shape or pattern at the point of crossing. To calculate the number of crossings, we need to find the cotangent of half the angle of crossing (α/2).

The reason we take half the angle of crossing is that the crossing occurs on both sides of the center line, and by considering only half the angle, we can count the crossings on one side and multiply it by 2 to get the total number of crossings.

The cotangent of an angle is defined as the ratio of the adjacent side to the opposite side of a right triangle. In this case, the adjacent side represents the distance between the center line and the line of the shape or pattern at the point of crossing, while the opposite side represents the distance between two consecutive crossings.

By taking the cotangent of half the angle of crossing (α/2), we can calculate the number of crossings per unit length along the center line. Since we are interested in the total number of crossings, we need to multiply this value by the total length of the shape or pattern.

Hence, the correct answer is option 'A' - 1/2 cot(α/2), which accurately represents the number of crossings 'N' according to the center line method.