In geometric modeling and processing, computer graphics and computer vision, smooth surfaces are approximated by discrete triangular meshes reconstructed from sample points on the surface. A fundamental problem is to design rigorous algorithms to guarantee the geometric approximation accuracy by controlling the sampling density. This theoretic work gives explicit formula to the bounds of Hausdorff distance, normal distance and Riemannian metric distortion between the smooth surface and the discrete mesh in terms of principle curvature and the radii of geodesic circum-circle of the triangles. These formula can be directly applied to design sampling density for data acquisition and surface reconstructions. Furthermore, we prove the meshes induced from the Delaunay triangulations of the dense samples on a smooth surface are convergent to the smooth surface under both Hausdorff distance and normal fields. The Riemannian metrics and the Laplace-Beltrami operators on the meshes are also convergent. These theoretic results lay down the theoretic foundation for a broad class of reconstruction and approximation algorithms in geometric modeling and processing.