Moving a shape around the earth while maintaining the proper projection

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What is the transformation needed to be taken on every data point of a shape to rotate that shape around the globe.

I'm trying to figure out the math behind this kind of transformation so I can create something similar to this using leaflet and d3 https://gmaps-samples.googlecode.com/svn/trunk/poly/puzzledrag.html

Edit: To be clear, I am looking to achieve this using leaflet, D3, and javascript not desktop GIS software. My end product will a function that takes a point (lat,lng) shifts it by an amount (shift_lat,shift_lng) and returns (new_lat,new_lng)



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