Matrix-valued subdivision schemes for generating surfaces with extraordinary vertices

Subdivision templates of numerical values are replaced by templates of matrices in this paper to allow the introduction of shape control parameters for the feasibility of achieving desirable geometric shapes at those points on the subdivision surfaces that correspond to extraordinary control vertices. Formulation of the matrix-valued subdivision surface is derived. Based on refinable bivariate spline function vectors for matrix-valued subdivisions, the notion of characteristic map introduced by Reif is extended from (scalar) surface subdivisions to matrix-valued subdivisions. The Cand C-continuity of Reif and Prautzsch for matrix-valued subdivisions are discussed. To illustrate the general theory, the smoothness of matrix-valued triangular subdivision schemes for extraordinary vertices with valences 3 and 4 is analyzed. The issue of effective choices of the shape control parameters will also be discussed in this paper.