#### Document Type

Dissertation

#### Degree

Doctor of Philosophy

#### Major

Applied Mathematics

#### Date of Defense

11-17-2011

#### Graduate Advisor

Qingtang Jiang

#### Committee

Charles K. Chui

Wenjie He

Henry Kang

#### Abstract

Surface subdivision schemes are used in computer graphics to generate visually smooth surfaces of arbitrary topology. Applications in computer graphics utilize surface normals and curvature. In this paper, formulas are obtained for the first and second partial derivatives of limit surfaces formed using 1-ring subdivision schemes that have 2 by 2 matrix-valued masks. Consequently, surface normals, and Gaussian and mean curvatures can be derived. Both quadrilateral and triangular schemes are considered and for each scheme both interpolatory and approximating schemes are examined. In each case, we look at both extraordinary and regular vertices. Every 3-D vertex of the refinement polyhedrons also has what is called a corresponding “shape vertex.” The partial derivative formulas consist of linear combinations of surrounding polyhedron vertices as well as their corresponding shape vertices. We are able to derive detailed information on the matrix-valued masks and about the left eigenvectors of the (regular) subdivision matrix. Local parameterizations are done using these left eigenvectors and final formulas for partial derivatives are obtained after we secure detailed information about right eigenvectors of the subdivision matrix. Using specific subdivision schemes, unit normals so obtained are displayed. Also, formulas for initial shape vertices are postulated using discrete unit normals to our original polyhedron. These formulas are tested for reasonableness on surfaces using specific subdivision schemes. Obtaining a specified unit normal at a surface point is examined by changing only these shape vertices. We then describe two applications involving surface normals in the field of computer graphics that can use our results.

#### Recommended Citation

Smith, James J., "Geometric Structures on Matrix-valued Subdivision Schemes" (2011). *Dissertations*. 408.

https://irl.umsl.edu/dissertation/408