Curve Shape Modification and Fairness Evaluation

A method to generate a quintic NURBS curve which passes through the given points is described. In this case, there are four more equations than there are positions of the control points. Therefore, four gradients which are the first derivative of a NURBS equation are assigned to the given points. In addition to this method, another method to generate a quintic NURBS curve which passes through the given points and which has the first derivative at these given points is described. In this case, a linear system will be underdetermined, determined or overdetermined depending on the number of given points with gradients. A method to modify NURBS curve shape according to the specified radius of curvature distribution to realize an aesthetically pleasing freeform curve is described. The differences between the NURBS curve radius of curvature and the specified radius of curvature is minimized by introducing the least-squares method. A criterion for a fair curve is proposed. Evaluation whether the designed curve is fair or not is accomplished by a comparison of the designed curve to a curve whose radius of curvature is monotone. The radius of curvature is specified by linear, quadratic, and cubic function using the least-squares method. A curve whose radius of curvature is reshaped by one of these algebraic functions is considered as a fair curve. The curvature vector of the curve is used to evaluate the fairness. The comparison of unit curvature vectors is used to evaluate the directional similarity of the curve. The comparison of the curvature is used to evaluate the similarity of the magnitude of curvature vectors. If the directional similarity of the designed curve is close to the fair curve, and also the similarity of the curvature is close to the fair curve, the designed curve can be judged as a fair curve.