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I have three points in space, which cannot move relative to one another, and create a reference plane.

There is a forth point, that lays off/on the plane (off will be more general solution, on will be a private case I guess).

How can I use the information I just described to predict where the point off/on the plane will lie when the plane moves? The point is constrained by the original relationship.

For example: $P_1$, $P_2$, $P_3$ define a plane and they are respectively $(1,1,5) , (0,1,5) , (-1,0,5)$.
The fourth point $P$ is $(-5,2,5)$ on the plane defined by $P_{1,2,3}$. What would be the coordinates of $P$ when the plane moves (arbitrary rotation in space) and $P_{1,2,3}$ have new values. In the example all points are on the same $z$ plane $(5)$ in the initial reference state.

Thanks!

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    Are only rotations allowed? When you say 'rotation' you mean rotations around any axis, right? Are the old values of $P_i$ mapped onto the new ones (meaning that the angles of the triangle $P_1P_2P_3$ are preserved)?2012-08-14
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    rotation could be any rotation, not just around specific axis. the old values are mapped. imagine a tool with 4 significant points that travels freely in the 3d space.2012-08-14

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