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In this paper, we present a new minimal solution for the extrinsic calibration of a 2D light detection and ranging (LIDAR) sensor and a perspective camera. This problem is formulated as registering three planes and the corresponding coplanar lines. All existing algorithms solve this problem by its geometric structure in the original or the dual 3D space. In contrast, our algorithm directly exploits the algebraic structure of the polynomial system to resolve this problem. This new algorithm is more abstract, however, and provides a more broadly applicable method to other problems that need to handle the similar polynomial system. The rotation matrix is estimated first. Then, the translation vector can be calculated by solving a system of three linear equations. Although the new approach is conceptually simple, it has 720 different versions caused by different permutations of variables. This results in different computational orders and affects the numerical behavior of the algorithm. A simple heuristic scheme is proposed to select the permutation of variables that yields numerically stable computational order with respect to the given input. Simulation and experimental results show that the proposed algorithm outperforms the existing state-of-the-art algorithms in terms of accuracy and numerical stability.