has been changed, which should not happen in reality. The reason the core algorithm of R1PP$n$P may generate points with different left-right-handed shape is that its rotation estimation equation does not constrict $\det ({\bf{R}}) = 1$.

In practice we found it not appropriate to constrain $\det ({\bf{R}}) = 1$ from the beginning of the algorithm. Instead, we allow the iteration process to approach the mirror-image form. This is because we found that, with the constrain $\det ({\bf{R}}) = 1$ from the beginning, the algorithm has many types of local minima and they are unpredictable. However, without this constrain, the convergence direction of the core algorithm becomes predictable, with only two types of convergence. The algorithm may reach the global optimal result directly or the approximate mirror-image form. For the latter case, the estimated $\det ({\bf{R}}) = -1$.

Hence, according to the above analysis, we propose the local minima avoidance mechanism. The algorithm begins with a random initial value and control point. When the algorithm converges to a result with $\det ({\bf{R}}) = -1$, we perform a mirror flip by

$${\lambda _{i,\rm{new}}} = 1/{\lambda _{i,\rm{old}}},{\kern 5pt} i = 1,...,N {\kern 5pt} {\rm{and}} {\kern 5pt} i\ne o.
\label{eq_lambdaflip}$$

Control Point $o$ Selection
