DR. JAN HADENFELD
		Distance Tolerance Up to now we did not take care of any distance tolerance. But it is often necessary to fulfill a prescribed tolerance $\delta$ between the old and the smoothed curve. So, we have to take care of the constraint $$\max \{ \vert {\bf x}(t) - \widetilde{\bf x}(t) \vert \; : \;t \in [t_{k-1},t_{n+1}] \} \le \delta$$ (19) in each step. But this constraint also leads to a nonlinear-problem and resulting we do use $$\vert {\bf d}_r - \widetilde{\bf d}_r \vert \le \delta$$ (20) as an upper bound for (19) (c.f. [30]). Let $\widetilde{\bf d}_r^{\; \ast}$ be the control point which minimizes the fairness functional under the constraint (20). Two cases have to be distinguished: firstly, the new control point $\widetilde{\bf d}_r$ satisfies the constraint (20). In this case nothing else has to be done (see Fig. 1 left). Secondly, the constraint is not fulfilled. In this case we are searching for a new location $\widetilde{\bf d}_r^{\; \ast}$. Here we can use the fact that the isolines of the energy integral are in the planar case concentric circles (spheres in ${\Bbb R}^3$) with center $\widetilde{\bf d}_r$. Then the new point is determined by (see Fig. 1 right) $$\widetilde{\bf d}_r^{\; \ast} = {\bf d}_r + \delta \cdot \fr... ...r} {\displaystyle \vert \widetilde{\bf d}_r - {\bf d}_r \vert}$$ (21) Figure: The distance tolerance is fulfilled (left) or not (right).