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Using QGIS, I want to take a EPSG 4326 (WGS84 LatLong) shapefile of Countries the world and "rotate" it such that Australia or some other area is at the north pole. The new map would now still be in Lat Long, only the positions of the countries will have changed/distorted appropriately. How the heck would I do that? I'll want to run this through modelbuilder in order to move other data such as rivers, lakes, etc through the same rotation (LatLong translation?).

enter image description here

The attached photo roughly describes what I'm trying to do. In the left globe, north is up (not modified). In the right globe, I've rotated the UVW mapping around the sphere arbitrarily. Now, I'd like to "unwrap" this back to a EPSG 4326 (WGS84 LatLong) projection with the landforms in their new positions.

How can this be done?

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The quick way to do this is with the Vector Bender plugin. It's similar to the Spatial Adjustment toolset in ArcMap, which is like georeferencing rasters but for vectors. You can rotate, which is what you want, or reshape using reference points.

If you don't see it in Plugins library you can download it here.

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