How can I make a material that merges from one object onto another? In Vue, I can easily make a material that merges between different objects. (That Vue material setting is called "World Standard" if it matters.) But Vue costs lots of money, and Blender is a free software for free people who run for the horizon in the summer evening's sunset to nice background music. That's why I want to know how to do this in Blender.

Please look at the below illustration from Vue:
As @Serge has commented, you can get the position of the shading-point in measured in World Space by using the Position output of the 'Geometry' input node. 

You can also get the point measured in any other object's space (carrying that object's transform,) by using the Object output of 'Texture Coordinate' node, and referring to the other object in the slot provided at the bottom of the node.