I have two polygons layers, one (grey) is Roads, other (blue) is Fields. Can I calculate distances between Fields (by field unique ID number) using Roads layer as distance measurement? What tools or extensions should I use?
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First, I think you will need to convert your road polygons to road centerlines, for which there is a Collapse Dual Lines To Centerline (Cartography) tool available, but it requires an Advanced level license.
Once you have your road network as line features ...
The ArcGIS for Desktop extension designed to model road networks is Network Analyst, but it can also do a lot more, and so may be overkill if your requirements are no greater than those that you have already described:
With the ArcGIS Network Analyst extension, you can answer questions like the following:
- What is the quickest way to get from point A to point B?
- Which houses are within five minutes of a fire station?
- What market areas does a business cover?
- A person wants to visit a store. Which branch should the potential customer visit to minimize travel time?
- Which ambulances or patrol cars can respond quickest to an incident?
- How can a fleet of delivery or service vehicles improve customer service and minimize transportation costs?
- Where can a business open a store to maximize market share?
- If a company has to downsize, which stores should it close to maintain the most overall demand?
- What are live or historical traffic conditions like, and how do they affect my network analysis results?
Not really an answer but I don't think that I can add an image into a comment. If you are going to use tractors, trucks and harvesters for driving from field to field it is most common to define also "gates" which are the points where you can really enter the fields. You must also define if the gate or the centroid of the field is better for distance calculations. Gate is often good if you do work analysis, tractors are driving so much faster on the road than when pulling a plough on the field.