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I have a polygon in QGIS 2.14 and want to generate a specified number of coordinate points within that polygon, with a minimum distance apart. The fTools Random Points tool doesn't give this option, and neither does the "Point sampling" analysis tool in the Point Sampling plugin (or maybe I don't know how to use it properly). Is there a way to do this in QGIS? I know it's quite straightforward in the Random Points tool in ArcGIS.

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You can use Random points inside polygons (fixed) from Processing Toolbox. If you don't see Processing Toolbox panel, activate it from menu Processing --> Toolbox. For distances in meters you have to use projected CRS (e.g some UTM) for more information about CRS see QGIS Doucumentation

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Point sampling tool that you mentioned, is good for retrieving values from raster layers.

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