Wind speed calculation

I want to know the impact of wind on endurance.

So we have something like this:

This gives us 8 cases of wind direction:

From this we can infer that going to the center from one of the 8 directions.

So an overall flight path would look like this:

If each grid cell is a distance (s) in size, each time a diagonal transition is made the distance would be (sqrt{2} s). As an aircraft enters each cell, they encounter the wind-star as shown above with the expected value of the wind facing one of the eight possible directions. In general, if (W) is the wind velocity in direction (alpha), from grid direction (g in [1,8]) we have to find the expected value for the wind in direction (cos(alpha_{g,w})), by integrating over all values of (w).

[
V_{g,w} = sum_{1..12} cos(alpha_{g,w_{i}}) dot P_i
]

With these bins:

h3. Wind Directions ( w_{i} )
|Wind Direction|Angle|
|1|0|
|2|30|
|3|60|
|4|90|
|5|120|
|6|150|
|7|180|
|8|210|
|9|240|
|10|270|
|11|300|
|12|330|

h3. Grid Direction (g)
|Grid Direction||
|1|315|
|2|270|
|3|225|
|4|0|
|5|180|
|6|45|
|7|90|
|8|135|

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