we denote the camera frame as $c$ and the world frame as $w$. For point $i$, without taking into account the distortion, the perspective projection equations are employed to describe the pinhole camera model,

$${u_i} = f\frac{{x_i^c}}{{z_i^c}},{v_i} = f\frac{{y_i^c}}{{z_i^c}},
\label{eq_1}$$

where $f$ is the camera focal length, ${\bf{x}}_i = {[{u_i},{v_i},f]^T}$ is the image homogeneous coordinate in pixels, and ${\bf{X}}_i^c = [x_i^c,y_i^c,z_i^c]^T$ is the real-world coordinate with respect to the camera frame.

According to ,

$${\bf{X}}_i^c = \lambda_i^* {\bf{x}}_i,
\label{eq_2}$$

where $\lambda_i^* = z_i^c / f$ is the normalized depth of point $i$. indicates that an object point lies on the straight line of sight of the related image point.

The relationship between the camera and world frame coordinate of point $i$ is $${\bf{X}}_i^c = {\bf{R}}{\bf{X}}_i^w + {\bf{t}},
\label{eq_3}$$

where ${\bf{R}}\in SO(3)$ is the rotation matrix and ${\bf{t}}\in R^3$ is the translation vector. ${\bf{R}}$ and ${\bf{t}}$ are the variables that need to be estimated in the P$n$P problem.

Similarly to the translation elimination method used in works [@kneip2013using][@lee2015minimal], with two points $i$ and $o$,

$${\bf{X}}_i^c - {\bf{X}}_{o}^c = {\bf{R}}({\bf{X}}_i^w - {\bf{X}}_o^w),{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} i \ne o.
\label{eq_4}$$

In the proposed R1PP$n$P algorithm, 