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I have one zone system and two layers. The 1st layer is the geographical centroids of the zone system (each zone has 1 geographical centroid), while the 2nd layer displays the rail stations. I would like to calculate and display the distance between the centroids and the rail stations. Ideally, I want each station to be connected to the centroid of the zone that it belongs (see pic attached).

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In essence, it is the shortest distance, but in some cases, one rail station might be closer to a geographical centroid of another zone and QGIS will give me this distance, which I don't want. In addition, one zone might have more than 1 rail stations. In that case, I want only the shortest distance.

Any idea on how to perform this task?

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You will need to code this function down yourself. If coordinates for both centroid (cX,cY) and station (sX,sY) are known - you may use following function to compute 2D distance between two points:

 Math.Sqrt((sX - cX) ^ 2 + (sY - cY) ^ 2)

If you are new to coding, I suggest you use Python language for this (please see QGIS - Getting Started With Python Programming).

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