DR. JAN HADENFELD
		Distance tolerance If we replace the control point $\bar{\mbox{\bf d}}_r$ by $\widetilde{\mbox{\bf d}}_r$ as computed above in (17) then the originally posed constraint (13), controlling the prescribed $\delta$-distance between $\mbox{\bf x}$ and $\widetilde{\mbox{\bf x}}$, is not yet taken into consideration. In order to overcome this problem we will use the very simple condition $$\vert {\mbox{\bf d}}_r - \widetilde{\mbox{\bf d}}_r \vert \le \delta$$ (23) which guarantees by help of the convex-hull property of B-spline curves that (13) is fulfilled even if the local fairing step is repeated several times at different control points. Condition (23) represents a good upper bound for the actual pointwise error in (13), at least, if the polynomial degree of the spline is small; c.f. (de Boor, 1978) and (Schaback, 1993) where some alternative bounds are discussed. Now, with respect to (23) we have to distinguish two cases. The first one is obvious, namely condition (23) is already fulfilled. Then $\widetilde{\mbox{\bf d}}_r$ is lying within the $\delta$-sphere with the centre ${\mbox{\bf d}}_r$. In the second case the new control point is situated outside this sphere and we are therefore interested in that alternative position $\widetilde{\mbox{\bf d}}_r^{\; \ast}$ on the boundary of the distance sphere in which the value of the energy integral $E_l(\widetilde{\mbox{\bf d}}_r^{\; \ast})$ is as small as possible. Here, we can use the fact already mentioned that the isolines of the energy integral are concentric spheres with centre $\widetilde{\mbox{\bf d}}_r$ in order to compute the location $\widetilde{\mbox{\bf d}}_r^{\; \ast}$ by $$\widetilde{\mbox{\bf d}}_r^{\; \ast} = {\mbox{\bf d}}_r + \d... ...yle \vert \widetilde{\mbox{\bf d}}_r - {\mbox{\bf d}}_r \vert}$$ (24) or equivalently as the intersection point of the connecting line from ${\mbox{\bf d}}_r$ to $\widetilde{\mbox{\bf d}}_r$ and the $\delta$-sphere around ${\mbox{\bf d}}_r$. In Fig. 1 this geometrical situation is illustrated in the planar case. Figure 1: Satisfying the $\delta$-distance constraint.