Appendix B Surface Algorithms

This appendix describes the mathematical basis of the Maxsurf models, and will be of interest to users who wish to write software to recreate Maxsurf surfaces.

Maxsurf constructs its shapes using the Rational B-spline formulation as described by the IGES standard. Maxsurf uses a uniform knot vector. However, non-uniform knot vectors are permissible.

B-spline curves and surfaces are dependent on a set of basis functions, which define the influence of each control point at any given point on the curve. The basis functions are determined by a vector of knots T where T = {t₀,...,t_i,t_i+1,...,t_m} and m = number of control points + order of the curve. The parameter t varies from zero to n – k + 2. In general, the B-spline basis function on a curve of degree p (order k = p + 1) is given by

                            1 if t_i £ t < t_i+1 and t_i < t_i+1

     N_i,0((t) =

                            0 otherwise

     N_i,p(t) = t - ti   N_i,p-1(t)   +       t_i+p+1 - t    N_i+1,p-1(t)                                

                             t_i+p-t_i                                                  t_i+p+1 - t_i+1                 assuming  0/0=0.

The elements of the open uniform knot vector, t_i , are given by

     t_i = 0                1 £ i £ k

     t_i = i - k            k + 1 £  i  £ n + 1

     t_i = n – k + 2    n + 2 £  i  £ n + k + 1

where there are n+1 control points on the curve, and k  is the order of the curve      

(k = p + 1).

A rational B-spline surface is given by

                                        _{m        n}

                                        S  S  N_i,p(u)  N_j,q(v)  W_ij P_ij

                                        ^{i=0      j=0}

     S(u,v) =           ______________________   _

                             _{m        n}

                            S  S  N_i,p(u) N_j,q(v) W_ij

                            ^{i=0      j=0}

where S(u,v) is a point on the B-spline surface at parametric coordinates u,v,

N_i,p(u) and N_j,q(v) are the B-spline basis functions, W_ij  are the rational weighting values, and P_ij are the control point positions.