In this paper, we proposed a feature extraction and simplification algorithm for three-dimensional (3D) point cloud based on piecewise linear Morse theory to solve problems in difficultly deleting pseudo feature points, and generating noise characteristic lines after simplification of existing 3D point cloud topological feature extraction algorithms based on Morse theory. First, we calculated the function index to extract the feature points. Regarding the triangles comprising each feature points as a collection, we calculated the dot product of the normal vectors of two adjacent triangles in the collection. Herein, the dot product maximum was considered the weight of the point, and the appropriate threshold was set to remove the pseudo feature points. Second, feature lines were produced by connecting feature points to complete the construction of descending Morse complex. Finally, the persistence and the retention index considering the actual retention value of the feature lines were calculated, respectively, by choosing the function index value of different points on the feature lines. With both indexes, the origin persistence method can be improved to attain a new feature line measurement index and the simplification and expression of 3D point cloud topological feature can be accomplished. The results show that, compared with other algorithms, the proposed feature extraction algorithm significantly reduces pseudo feature points. Furthermore, the time efficiency of the proposed algorithm for constructing the descending Morse complex increases by 70.37%, and the point cloud compression rate increases 22.48%. Additionally, the new importance measurement method for feature lines attains more concise, continuous, and structural complete feature lines, and the feature extraction and simplification processes show strong anti-noise performance.