Subdivision schemes are the most effective and efficient tools to design, generate, edit, and display 3-dimensional surfaces. In animation movie production, the subdivision approach is the preferred way to represent everything that moves. The purpose of this thesis is to develop the theoretical aspects and to construct explicit matrix-valued coefficient stencils for the recently introduced 5 and 7 subdivisions. While the 5 rule was first proposed as a hierarchical sampling scheme over a regular mesh, the objective of the triangular 7 subdivision is for potential applications to pyramid algorithm design more for human vision modeling and for representing structures of very large carbon molecules in chemistry.;The most well-known subdivision schemes are the Loop and Catmull-Clack schemes. But they are not interpolatory. Interpolating schemes are necessary for many applications, such as reversed engineering, where the data points are used as control vertices, and therefore must stay unchanged and lie on the final subdivision surfaces. However, in the construction of any interpolating (scalar-valued) subdivision scheme, we must face the dilemma in that subdivision templates with undesirable large size are needed to achieve subdivision surfaces of any higher order of smoothness. Such subdivision templates are in general too large to be useful in practice, particularly when they have to be adjusted near extraordinary vertices. For this reason, vector subdivisions for surface design were recently studied.;The main results in this thesis include the introduction of matrix-valued coefficient stencils to construct both C2 approximating and C2 interpolatory 5 and 7 subdivision schemes, outline of a constructive procedure with demonstrative examples, and pseudo-codes for computer implementation.
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