Warp polygons/lines from distortion grid, or similar

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Anyone have any ideas how I might go about warping polygons or line based on point based distortion map?

Specifically, let's say I have a (non-regular) point data set. I create a clone of the data set and translate points in a non-regular fashion. Is there a way to use the difference of the two layers to warp a related line or polygon dataset? It doesn't need to be "accurate" (whatever that means in this instance), just give a rough indication of what the underlying topology looks like giving the applied distortion.

Using either PostGIS, QGIS, Shapely, GEOS etc. Or if you have an Arc* version I can't probably make that work too.



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