The control of polygons of b-spline subdivision curves are usually refined uniformly using a technique called knot-doubling. This uniform refinement approach would perform unnecessary subdivision steps on portions already close to the limit curve enough and, consequently, cause unnecessary (exponential) increase on the number of line segments in the refined polygons. This paper overcomes this problem by proposing a local refinement technique for the control polygons of B-spline subdivision curves. Local refinement is achieved by selectively inserting new knots at midpoints of knot intervals. Efficient adaptive subdivision can be easily realized based on the new technique.