To appear in Computer - Aided Design Revised June 18 , 2008

Both the 4-point and the uniform cubic B-spline subdivisions double the number of vertices of a closed-loop polygon P and produce sequences of vertices fj and bj respectively. We study the J-spline subdivision scheme Js, introduced by Maillot and Stam, which blends these two methods to produce vertices of the form vj=(1–s)fj+sbj. Iterative applications of Js yield a family of limit curves, the shape of which is parameterized by s. They include four-point subdivision curves (J0), uniform cubic B-spline curves (J1), and uniform quintic B-spline curves (J1.5). We show that the limit curve is at least C when –1.7≤s≤5.8, C when 0<s<4, C when 1<s≤2.8, and C when s=3/2, even though 4-point yields only C curves and cubic B-spline yields only C curves. We generalize the Js scheme to a two-parameter family Ja,b and propose data-dependent and data-independent solutions for computing values of parameters a and b that minimize various objective functions (distance to the control vertices, deviation from the control polygon, change in surface area, and popping when switching levels of subdivision in multi-resolution rendering). We extend the J-spline subdivision to open curves and to a smooth surface subdivision scheme for quad-meshes with arbitrary connectivity.