Abstract: Since T-splines cannot represent hyperbolic spline surfaces exactly,this paper presents a kind of spline surfaces,called non-uniform algebraic hyperbolic T-spline surfaces(NUAH T-splines for short) of odd bi-degree.The NUAH T-splines are defined by applying the T-spline framework to the non-uniform algebraic hyperbolic B-spline surfaces(NUAH B-spline surfaces).Based on the knot insertion of NUAH B-splines,a local refinement algorithm for NUAH T-splines of odd bi-degree is shown.This paper proves that,for any NUAH T-spline of odd bi-degree,the linear independence of its blending functions can be determined by computing the rank of the NUAH T-spline-to-NUAH B-spline transformation matrix.Finally,the examples verify the effectiveness of the local refinement algorithm of NUAH T-splines.