DR. JAN HADENFELD
		Ranking number In the two previous subsections we have learnt how to change the curve locally by considering a $\delta$-distance. Now, the question is: at which point should we fair a given B-spline curve? Here, a natural answer is to fair the curve at that control point $\bar{\mbox{\bf d}}_r$ where the largest improvement of the energy integral is to be expected. Thus, we introduce as a local fairness criterion the following positive number $$z_r = E_l(\bar{\mbox{\bf d}}_r) - E_l(\widetilde{\mbox{\bf d}}_r) \ge 0$$ (25) or in more detail with help of the notations (10) and (11) $$z_r = \int_a^b \left( \bar{\mbox{\bf x}}^{(l)}(t) \right)^2 ... ...eft( \widetilde{\mbox{\bf x}}^{(l)}(t) \right)^2 \; dt \;\;\;.$$ (26) After some calculations, one verifies directly that (26) can simply be expressed as $$z_r = \left(\bar{\mbox{\bf d}}_r - \widetilde{\mbox{\bf d}}_... ...ight)^2 \cdot \int_a^b \left( N_{r,k}^{(l)}(t) \right)^2 \; dt$$ (27) and we recognize that the local criterion (ranking number) is just the squared improvement of the control point $\bar{\mbox{\bf d}}_r$ weighted by the integral of the squared and $l$-times differentiated basis function $N_{r,k}(t)$. Finally, following the performances of the previous subsection we can also modify (27) in order to take care of the distance tolerance $\delta$ already in the ranking number by using $$z_r^{\; \ast} = \left(\bar{\mbox{\bf d}}_r - \widetilde{\mbo... ...ight)^2 \cdot \int_a^b \left( N_{r,k}^{(l)}(t) \right)^2 \; dt$$ (28) if $\vert {\mbox{\bf d}}_r - \widetilde{\mbox{\bf d}}_r \vert > \delta$, although (28) is not equivalent to $\;\;E_l(\bar{\mbox{\bf d}}_r) - E_l(\widetilde{\mbox{\bf d}}_r^{\; \ast})$.