Using the projected curve of surface mesh boundary as parameter domain border, linear mapping parameterization with natural boundary is realized. A fast algorithm for least squares fitting plane of vertices in the mesh boundary is proposed. After the mesh boundary is projected onto the fitting plane, low-pass filtering is adopted to eliminate crossovers, sharp corners and cavities in the projected curve and convert it into an eligible convex parameter domain boundary. In order to facilitate quantitative evaluations of parameterization schemes, three distortion-measuring formulae are presented. Using the projected curve of surface mesh boundary as parameter domain border, linear mapping parameterization with natural boundary is realized. A fast algorithm for least squares fitting plane of vertices in the mesh boundary is proposed. After the mesh boundary is projected onto the fitting plane, low-pass filtering is adopted to eliminate crossovers, sharp corners and cavities in the projected curve and convert it into an eligible convex parameter domain boundary. In order to facilitate quantitative evaluations of parameterization schemes, three distortion-measuring formulae are presented.