J J -splines
Both the 4-point and the uniform cubic B B -spline subdivisions double the number of vertices of a closed-loop polygon kP k P and produce sequences of vertices fj f j and bj b j respectively. We study the J J -spline subdivision scheme Js J s , introduced by Maillot and Stam, which blends these two methods to produce vertices of the form vj=(1?s)fj+sbj v j = ( 1 ? s ) f j + s b j . Iterative applications of Js J s yield a family of limit curves, the shape of which is parameterized by s. They include four-point subdivision curves (J0 J 0 ), uniform cubic B B -spline curves (J1 J 1 ), and uniform quintic B B -spline curves (J1.5 J 1.5 ). We show that the limit curve is at least C1 C 1 when ?1.7?s?5.8 ? 1.7 ? s ? 5.8 , C2 C 2 when 0<s<4 0 < s < 4 , C3 C 3 when 1<s?2.8 1 < s ? 2.8 , and C4 C 4 when s=3/2 s = 3 / 2 , even though a 4-point yields only C1 C 1 curves and a cubic B B -spline yields only C2 C 2 curves. We generalize the Js J s scheme to a two-parameter family Ja,b J a , b and propose data-dependent and data-independent solutions for computing values of parameters a a and b 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 J -spline subdivision to open curves and to a smooth surface subdivision scheme for quad-meshes with arbitrary connectivity.
subdivision curves four-point surfaces continuity b-splines interpolation fitting area preservation