NUAT T-splines of odd bi-degree and local refinement

This paper presents a new kind of spline surfaces, named non-uniform algebraic-trigonometric T-spline surfaces (NUAT T-splines for short) of odd bi-degree. The NUAT T-spline surfaces are defined by applying the T-spline framework to the non-uniform algebraic-trigonometric B-spline surfaces (NUAT B-spline surfaces). Based on the knot insertion algorithm of the NUAT B-splines, a local refinement algorithm for the NUAT T-splines is given. This algorithm guarantees that the resulting control grid is a T-mesh as the original one. Finally, we prove that, for any NUAT T-spline of odd bi-degree, the linear independence of its blending functions can be determined by computing the rank of the NUAT T-spline-to-NUAT B-spline transformation matrix.