We have two line segments specified by a pair of points $(b_1,t_1)$, and $(b_2,t_2)$. We want to find the transformation between the two line segments. Define a unit vector $z_1$ from $b_1$ and pointing towards $t_1$. Similarly, define $z_2$. Then, I want to find some translation vector and rotation matrix, such that any point in $z_2$ can be expressed in terms from $z_1$. I understand that the problem is slightly ill-posed because there can be multiple solutions.
Define points and get rotation matrices:
b1 = {1, 0, 1}; t1 = {1, 2, 1}; b2 = {0, 1, 0}; t2 = {1, 1, 0}; z1 = (t1 - b1)/Norm[(t1 - b1)]; z2 = (t2 - b2)/Norm[(t2 - b2)]; Rot1 = RotationMatrix[{{0, 0, 1}, z1}] Rot2 = RotationMatrix[{{0, 0, 1}, z2}] Get $4\times 4$ transformation matrices:
T1 = Join[MapThread[Append, {Rot1, b1}], {{0, 0, 0, 1}}] T2 = Join[MapThread[Append, {Rot2, b2}], {{0, 0, 0, 1}}] T21 = T2 . Inverse[T1] {r1, l1, r2, l2} = {0.1, Norm[b1 - t1], 0.1, Norm[b2 - t2]}; t21 = (T21 . {0, 0, 0, 1})[[1 ;; 3]]; The distances between the points don't hold. They should remain the same after transformation.
Norm[b2 - b1] Norm[t21] Position and orientation of line segments after transformation
Show[Graphics3D[{Cylinder[{{0, 0, 0}, {0, 0, l1}}, r1]}], Graphics3D[Cylinder[{t21, (T21 . {0, 0, l2, 1})[[1 ;; 3]]}, r2]], Boxed -> False] 
Position and orientation of line segments before transformation
Show[Graphics3D[{Cylinder[{b1, t1}, r1]}], Graphics3D[Cylinder[{b2, t2}, r1]], Boxed -> False] 
The two figures look different from each other. Also, distance between points are not maintained. Am I using RotationMatrix correctly? Also, I might be wrong about mathematical concept of transformation.
Edit(Clarification): Consider a coordinate system {1} such that $z_1$ (line segment 1) is aligned with the $z$-axis of the coordinate system {1} and origin at $b_1$. Similarly, let $z$-axis of coordinate system {2} align with $z_2$ (line segment 2) and origin at $b_2$. Then I want to express the second line segment in terms of the coordinate system {1} in the form of some translation and rotation.
