the points are collinear. However, for plane and line, there exist configurations which have several solutions. We call them as ambiguous configurations. The ambiguous configuration differs from the degenerate configuration. The ambiguous configuration has several solutions, but the degenerate configuration has infinite solutions. Formally, we have the following lemma for point-to-line correspondences. **Proof is in the supplementary material.**

\[lemma:line\] For any $n_l$ points and any $\left\{ {{{\bf{R}}_1},{{\bf{t}}_1}} \right\}$ and $\left\{ {{{\bf{R}}_2},{{\bf{t}}_2}} \right\}$, there exist $n_l$ lines to make $\left\{ {{{\bf{R}}_1},{{\bf{t}}_1}} \right\}$ and $\left\{ {{{\bf{R}}_2},{{\bf{t}}_2}} \right\}$ are exact solutions for the $n_l$ point-to-line correspondences.

Similarly, we have lemma \[lemma:plane\] for point-to-plane correspondences.

\[lemma:plane\] For any $n_{\pi}$ points and any $\left\{ {{{\bf{R}}_1},{{\bf{t}}_1}} \right\}$, $\left\{ {{{\bf{R}}_2},{{\bf{t}}_2}} \right\}$ and $\left\{ {{{\bf{R}}_3},{{\bf{t}}_3}} \right\}$, there exist $n_{\pi}$ planes to make $\left\{ {{{\bf{R}}_1},{{\bf{t}}_1}} \right\}$, $\left\{ {{{\bf{R}}_2},{{\bf{t}}_2}} \right\}$ and $\left\{ {{{\bf{R}}_3},{{\bf{t}}_3}} \right\}$ are exact solutions for the $n_{\pi}$ point-to-plane correspondences.

Finally, we have the following theorem for $n_l$ point-to-line and $n_{\pi}$ point-to-plane correspondences.

\[theorm:1\] For any $n_l$ points on lines and $n_{\pi}$ points on planes and any $\left\{ {{{\bf{R}}_1},{{\bf{t}}_1}} \right\}$ and $\left\{ {{{\bf{R}}_2},{{\bf{t}}_2}} \right\}$, there exist $n_l$ lines and $n_{\pi}$ planes to make $\left\{ {{{\bf{R}}_1},{{\bf{t}}_1}} \right\}$ and $\left\{ {{{\bf{R}}_2},{{\bf{t}}_2}} \right\}$ are exact solutions for the $n_l$ point-to-line and 