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Examples

The next given three examples are benchmark curves of the workshop in Lambrecht which have to be faired. In all cases only one control point on each side is fixed and the maximum perpendicular error (max_error) is given in relation to the maximal diagonal. All calculations had been done on a HP 9000/735 workstation.

In the first example the curve is a spatial one. To get a better impression the values of the curvature are not plotted as porcupines but as circles and the direction of the normals are visualized as lines1. The fairing result in the case of minimizing the second derivative is not of the same quality as in the case of the third derivative. The reason could be that by minimizing the third derivative this integral is the linearization of $(\dot{\kappa})^2 + \kappa^2 (\kappa^2 + \tau^2)$ and the torsion is also involved as well.

The second and third curves are nearly planar. Only the porcupines here are plotted to visualize the fairness of the curves.

Figure: The given B-spline curve named chine3, the faired one with $E_2$ (max_error=$0.19\%$, $0.19$ sec.) and $E_3$ (max_error=$0.15\%$, $0.19$ sec.).
\begin{figure} \centerline{\hfill \epsfxsize 40mm \epsfbox{chine_ori.ps} \hfill ... ...ox{chine_2.ps} \hfill \epsfxsize 40mm \epsfbox{chine_3.ps} \hfill}\end{figure}

Figure: The given B-spline curve named pr_krumm and the faired one (max_error=$0.20\%$, $0.09$ sec.).
\begin{figure} \centerline{\hfill \epsfxsize 65mm \epsfbox{pr_krumm_initial.ps} \hfill \epsfxsize 65mm \epsfbox{pr_krumm_1.0.ps} \hfill}\end{figure}

Figure: The given B-spline curve named np113 and the faired one (max_error=$0.30\%$, $0.06$ sec.).
\begin{figure} \centerline{\hfill \epsfxsize 65mm \epsfbox{np113_initial.ps} \hfill \epsfxsize 65mm \epsfbox{np113_1.0.ps} \hfill}\end{figure}