I am confused between 3D rigid affine (with scale, without shear) transformation and Lie group's sim(3) matrix.
3D affine should be $[sR|t; 0 1]$ \begin{matrix} s\cdot r_{11} & s\cdot r_{12} & s\cdot r_{13} & t_1\\ s\cdot r_{21} & s\cdot r_{22} & s\cdot r_{23} & t_2\\ s\cdot r_{31} & s\cdot r_{32} & s\cdot r_{33} & t_3\\ 0 & 0 & 0 & 1 \end{matrix}
we rotate, scale and then translate a 3D point using 3D affine transform. Here the translation is done after rotate and scale.
sim3, according to [1], the definition is $[{R} \vert t; 0 1/s]$
\begin{matrix} r_{11} & r_{12} & r_{13} & t_1\\ r_{21} & r_{22} & r_{23} & t_2\\ r_{31} & r_{32} & r_{33} & t_3\\ 0 & 0 & 0 & s^{-1} \end{matrix}
I have a two sets of 3D points. I used a c++ library to estimate the rigid transformation (with scale) between the points. How do I know if the answer is a 3D affine matrix or a sim3 matrix?
Can we have different matrices for a sim3 and affine transform that takes us from 3D points set-1 to set-2?
Is sim3, a way of representing the sequence rotate-translate-scale ?
Reference: [1] http://ethaneade.com/lie_groups.pdf
maybe this paper very be helpful. (I am not able to add as a comment, sorry)