How can I calculate the transformation of two 3D triangles?

Question

Given two triangle I have the transformation (three rotation followed by three translation)of both the triangles. How can I calculate the transformation between two triangles? A numerical example will help. Thanks in advance!

Answer 1

When rotating a rigid body into standard position, the three rotations are not one around each spatial axis. One approach is Euler angles. In any case, you would hope to have three single-axis rotation matrices that, when multiplied, give the whole rotation.

Given that this is true, if the matrices for triangle $ABC$ are $R_1,R_2,R_3$ and the translation is $T_1$, and for $DEF$ are $R_4,R_5,R_6,T_2$, and $A'B'C'$ is triangle $ABC$ in standard position (after the rotation/translation) you have $A'B'C'=T_1+R_3R_2R_1ABC=D'E'F'=T_2+R_6R_5R_4DEF$. Then $DEF=R_4^{-1}R_5^{-1}R_6^{-1}(T_1+R_3R_2R_1ABC-T_2)$