In this paper, a novel method of conformal parameterization for triangular meshes is presented. Firstly, based on geodesic on a mesh, an algorithm constructing local barycentric coordinates is proposed. Then, these local coordinates are merged via a linear system to form a global conformal parameterization of the mesh. The conformal mesh parameterization method here can be viewed as a development of the shape-preserving method proposed by M. S. Floater. It avoids error of locally approximating the so-called geodesic polar mapping and hence giving better results. Experimental results are given to illustrate the effectiveness of proposed methods.