Algebraic and Geometric Properties of a Family of Rational Curves

This paper consists of two components - an application part and a theoretical part, where the former targets the applications of computer aided geometric designs in generating parametric curves, and the latter focuses on the algebraic analysis of rational space curves. At the application level, we construct a family of rational space curves via quaternion products of two generating curves. At the theoretical level, we use algebraic methods to extract a $\mu$-basis for this family of curves, and describe a basis for a special submodule of the syzygy module in terms of a $\mu$-basis for the syzygy module of this family of curves. A commutative diagram is provided to summarize these results.