Particle Swarm Optimization Algorithm for B-spline Curve Approximation with Normal Constraint

If the knot vector and control points of a B-spline curve are variable, the B-spline curve approximation with normal constraint problem becomes a multidimensional, multivariate and highly nonlinear optimization problem with normal constraints, the conventional method of inverse equation system is difficult to obtain the optimal solution. Aiming at this kind of problem, a particle swarm optimization（PSO） method is introduced to solve the curve approximation problem with normal constraints. Firstly, the penalty function method is used to transform the constrained optimization problem into an unconstrained optimization problem. Secondly, a suitable fitness function which is closely related to both data points and normal constraints is constructed. Finally, PSO is applied to adjust the knot vector, and at the same time, the least square method is used to solve the optimal control points, do loop iteration until the best B-spline curve approximation is produced. By a comparison with existing methods, the superiority of the proposed method is highlighted. Test results show that this method is practical in solving the curve approximation problem with normal constraints.