3D correspondences with 6 effective constraints forms a minimal configuration. Table \[table:mini\_config\] lists the minimal configurations. We use the pattern Pt$i$L$j$Pl$k$ to represent $i$ point-to-point, $j$ point-to-line and $k$ point-to-plane correspondences. As one point-to-point, point-to-line and point-to-plane correspondence provide 3, 2 and 1 constraints, respectively, it is easy to verify that the configurations in Table \[table:mini\_config\] are minimal configurations, except for the case Pt$2$L$0$Pl$1$ (2 point-to-point and 1 point-to-plane correspondences). It seems that Pt$2$L$0$Pl$1$ has 7 constraints, as 2 point-to-point correspondences seemingly provide 6 constraints and 1 point-to-plane correspondence provides 1 constraint. However, the collinear configuration is degenerate for point-to-point correspondences [@arun1987least]. Two point-to-point correspondences form a collinear configuration. One additional constraint is required to recover $\bf{R}$ and $\bf{t}$. We propose an unified solution for these minimal cases.

        name        \#point   \#line   \#plane
  ---------------- --------- -------- ---------
   Pt$0$L$0$Pl$6$      0        0         6
   Pt$0$L$1$Pl$4$      0        1         4
   Pt$1$L$0$Pl$3$      1        0         3
   Pt$0$L$2$Pl$2$      0        2         2
   Pt$1$L$1$Pl$1$      1        1         1
   Pt$2$L$0$Pl$1$      2        0         1
   Pt$0$L$3$Pl$0$      0        3         0

  : Minimal configurations.[]{data-label="table:mini_config"}

Let us first consider the rational equation from (\[equ:9\]). Its numerator, ${\bf{k}}^T{\bf{v}}$, is a third order polynomial. Assume that we have 6 equations in the form of (\[equ:9\]). *This will lead to an 