Map points between 3D Coordinate systems

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I am trying to find a way to relate two 3D coordinate systems. I have 24 points for each system and found this, but it only works for 2D coordinate systems: http://msdn.microsoft.com/en-us/library/jj635757(v=vs.85).aspx

Is there any way to do something similar to the link above but with 3D Coordinate systems? I tried to follow the same strategy, but I'm stuck on which equations I'd need to be able to include z as well.