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I have two related tasks. How is it possible to solve them (or at the at least the first one, which is the principle task) in ArcGIS or QGIS ?

Task 1.

How to find a radius of a largest circle inscribed in a polygonal object (shapefile) with a random shape of the boundaries? Under such circle means a circle that touches the polygonal object at two or more points. (Any circle with larger radius will intersect sides of the polygonal object). Is it possible to solve this task in a vector?

If transform this task in a raster the question will be the following:

Task 2.

How to find for a group of cells of a raster (which represent the closed contour (a boundary) of a polygonal object from the first task which was put on a raster) a point (Latitude, Longitude) which lies at a maximum distance from all cells of this group and locates inside this closed contour? I need to determine the point’s coordinates and this “maximum distance”.

Are there any tools or scripts of this kind available in ArcGIS or QGIS?

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Although this already is a rather old post, I thought it might be useful to provide an "updated" answer. Updated in this context means that there is now (at least as of 05/2021) an open source solution that in my experience works very well.

QGIS3 has in its processing toolbox a tool called 'Pole of inaccessibility'. I work with land use polygons (of all imaginable shapes & sizes) and so far the tool has performed flawlessly for me. Yes, its rather slow & very memory intensive, but it provides the correct results. Below is a description of how the tool works.

"This algorithm calculates the pole of inaccessibility for a polygon layer, which is the most distant internal point from the boundary of the surface. This algorithm uses the 'polylabel' algorithm (Vladimir Agafonkin, 2016), which is an iterative approach guaranteed to find the true pole of inaccessibility within a specified tolerance (in layer units). More precise tolerances require more iterations and will take longer to calculate. The distance from the calculated pole to the polygon boundary will be stored as a new attribute in the output layer."

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  • An alternative way to create a pole of inaccessibility in QGIS i susing the expression pole_of_inaccessibility(geometry, tolerance), see: docs.qgis.org/3.16/en/docs/user_manual/working_with_vector/… - and gis.stackexchange.com/a/396974/88814 for a use case.
    – Babel
    Aug 18, 2021 at 7:25
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    @Babel: Good point 👍! Often its a lot handier to be able to use an expression rather than a separate tool, especially when creating processing chains. However, the tool also returns the distance between the POI and the nearest boundary, which in my case is handy (I want to create the maximum inscribed circle & its area to give me a hint on how many satellite imagery pixels the land use parcel covers). Aug 19, 2021 at 9:37

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