This paper describes a new computational method for creating a triangular and quadrilateral mesh of an arbitrary polygonal surface while controlling the anisotropy and directionality of the mesh. The input polygonal surface may include non-manifold edges and holes, and it can be either a closed polyhedron or an open polygonal surface. The method creates a mesh in two steps. In the first step, volumetric cells are packed - ellipsoidal bubbles for a triangular mesh, or rectangular solid bubbles for a quadrilateral mesh - on the input polygonal surface, then vertices are created at the centers of the bubbles and superimposed onto the input polygonal surface. In the second step, the original vertices of the input polygonal surface are deleted. The method is tolerant to the noise typically introduced by the measurement error of a laser range scanner, it can control element size and anisotropy precisely, and it creates high quality mesh elements. Applications of the proposed scheme include: reducing the number of elements of a mesh created by a laser range scanner, CT scanner, or MRI scanner; creating a surface mesh that can be a starting mesh for a tetrahedral or a hexahedral finite element mesh; and solution-adaptive anisotropic remeshing for finite element analysis.