According to this question,the correct result is Pr=R(Pl-T).R,T is the transformation from Caml to Camr.Pl is the position in Caml's camera coordinate system.Pr is the position in Camr's camera coordinate system.
In my opinion,if we know the transformation between two coordinate system,then it should be Pr=R*Pl+T.
I wonder to know where i was wrong.
Thanks a lot!
Basically, you’re getting the order of transformations backwards. In the left-hand camera’s coordinate system, the displacement vector $\mathbf T$ is just the coordinates of the right-hand camera center. To transform a point’s coordinates into the right-hand camera’s frame, you must first translate the origin to $\mathbf T$, then rotate: $\mathbf P_r = \mathtt R \left(\mathbf P_l-\mathbf T \right)$. You could rotate first, of course, but then the subsequent translation must be performed in this new rotated coordinate system, i.e., the displacement vector $\mathbf T$ must also be rotated. Algebraically, $\mathtt R \left(\mathbf P_l-\mathbf T \right) = \mathtt R \mathbf P_l-\mathtt R\mathbf T$.
I suspect that you may have been misled by the typical decomposition $\mathtt K \left[ \mathtt R \mid \mathbf t \right]$ of the camera projection matrix. In this decomposition, $\mathbf t$ is not the camera center $\tilde{\mathbf C}$ in world coordinates, but is in fact equal to $-\mathtt R \tilde{\mathbf C}$.