$ r_i=\|\boldsymbol{x}_i-\boldsymbol{x}_{i-1}\| $ | : The length of $ \mathcal{P}_i $ |
$ L= \sum\limits_{i=1}^Nr_i $ | : The total length of $ \mathcal{P} $ |
$ \boldsymbol{t}_i=(\boldsymbol{x}_i-\boldsymbol{x}_{i-1})/r_i $ | : The unit tangent vector on $ \mathcal{P}_i $ |
$ \boldsymbol{n}_i=-\boldsymbol{t}_i^\bot $ | : The outward unit normal vector on $ \mathcal{P}_i $ |
$ v_i $ | : A given representative normal velocity on $ \mathcal{P}_i $ |
$ \phi_i=\mathrm{sgn}(D_i)\arccos(\boldsymbol{t}_i\cdot\boldsymbol{t}_{i+1}) $ | : The angle between the adjacent edges $ \mathcal{P}_{i} $ and $ \mathcal{P}_{i+1} $ where $ D_i=\det(\boldsymbol{t}_i, \boldsymbol{t}_{i+1}) $ |
$ \boldsymbol{T}_i=(\boldsymbol{t}_i+\boldsymbol{t}_{i+1})/(2\mathsf{cos}_i) $ | : The unit tangent vector at $ \boldsymbol{x}_i $ where $ \mathsf{cos}_i=\cos(\phi_i/2)=\|(\boldsymbol{t}_i+\boldsymbol{t}_{i+1})/2\| $ |
$ \boldsymbol{N}_i=(\boldsymbol{n}_i+\boldsymbol{n}_{i+1})/(2\mathsf{cos}_i) $ | : The outward unit normal vector at $ \boldsymbol{x}_i $ |
$ V_i=(v_i+v_{i+1})/(2\mathsf{cos}_i) $ | : The normal velocity at $ \boldsymbol{x}_i $ |