This is a huge assumption that is almost always wrong.

Care to expand on that by showing us a practical example?

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.

This sends the X-Y plane to the X-Y plane

You're right. Rotation alone won't deal with parallel planes. I guess that's why I included:

(assuming they are not parallel)

The nice thing about rotation-only examples, as a starting point, is that they are easy to visualise, whilst getting away from thinking in terms of rotations occuring (only) around the orthogonal axis. Once you've mastered thinking in terms of rotations around arbitrary vectors, then applying a translation, once the points are in the same plane, comes fairly naturally. Reflections and sheers are somewhat harder to visualise (for me at least), but if you take simple steps, you get there.

Of course, if you're one of those that is comfortable just manipulating numbers in the abstract, then just applying someone elses formulae is all you need. But I find that without some grounding in visualising the way transforms combine it is nearly impossible to acquire a confidence in the results. And if you need to modify those formulae--to impose one or more constraints on the freedom of motion--then it is neigh impossible if you do not understand how the formulae work. YMMV.

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In the absence of evidence, opinion is indistinguishable from prejudice.

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I have just tried both cases my creating planes from each set of co-ordinates given.
Interestingly the first case worked fine by rotating points about the x-axis.
The second case failed (and was trapped) since these planes are parallel.

I guess that as soon as you have a translation of the points along the line, even the points on the line will move, because they "slide" up or down that line. Also, it's not necessarily clear to me why points on the line/intersection shouldn't move away from the line.