Let me give an example mapping the X-Y plane to the X-Z plane: 
(0, 0, 0) -> (0, 0, 0)
(1, 0, 0) -> (0, 0, 1)
(0, 1, 0) -> (-1, 0, 0)

[download]
Your approach would notice the 90 degree rotation in planes. But would assume that since (1, 0, 0) is in both planes that it gets mapped to itself.

For another interesting example consider the following mapping: 

(0, 0, 0) -> (1, 0, 0)
(1, 0, 0) -> (2, 0, 0)
(0, 1, 0) -> (1, 1, 0)

[download]
This sends the X-Y plane to the X-Y plane but no point remained still. Your approach has no idea how to handle it.

In reply to Re^4: (OT) moving points from one plane to another by tilly
in thread (OT) moving points from one plane to another by merrymonk

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