Abstract: In this paper, we present a new method for constructing five degree algebraic surface in homogeneous form using the cubic control surfaces over triangular meshes. The method has the features of quadratic precision, local shape control, efficient calculation, and straight forward geometric interpretation. Futher, intersections between such degree 5 algebraic surface and a class of proper planes can be a 4 degree algebraic curve and a line, and this simplies the parameterization of the algebraic surface.