I think my problem would be much easier to explain if I make you a sketch:
I've got two planes in $R^3~: π_1 ~~\text{and}~~ π_2$
$π_1~$ is defined by $~4~$ points : $~~ A = (a_1, a_2, a_3),~~ B = (b_1, b_2, b_3),~~ C = (c_1, c_2, c_3),~~ D=(d_1, d_2, d_3)~$
$π_2~$ is also defined by another $~4~$ points: $~~A_2 = (a_{21}, a_{22}, a_{23}),~~ B_2 = (b_{21}, b_{22}, b_{23}),~~ C_2 = (c_{21}, c_{22}, c_{23}),~~ D_2 = (d_{21}, d_{22}, d_{23})$
First what I need is to transform $~π_2~$ by changing the points $~C_2~$ and $~D_2~$. I need them to be the same value as $~C~$ and $~D~$.
Then I need to obtain points from that new $~π_2~$ plane with $~R_2~$ points as input. Like any $~2$D plane. I'm also drawing a sketch for that requirement because I think it will be easier to understand:
Anybody can help/save me? Thanks!