Evolution of a spiral-shaped polygonal curve by the crystalline curvature flow with a pinned tip

Figure 1.  Evolution of polygonal spiral. In the above figures, $ L_j $ and $ y_j $ denote the facets and vertices of the polygonal spiral, respectively

Figure 2.  Notations to the Wulff shape $ \mathcal{W}_\gamma $

Figure 3.  Example of positive convex (left) and positive concave(right) spiral for hexagonal $ \mathcal{W}_\gamma $

Figure 4.  Evolution of polygonal curves. The angle of the corners with a round curve is $ \theta_{j+1} - \theta_j $

Figure 5.  Examples of self-intersection between $ \Lambda_i (t)$ $ (solid line) and $ \Lambda_j (t) $ (dashed line). Note that dots means $ y_i (t) $ or $ y_j (t) $, which is a vertex belongs to $ \Lambda_i (t) $ or $ \Lambda_j (t) $, respectively

Figure 6.  The case of facet-vertex intersection with $ \Lambda_j (\bar{t}) \subset \mathcal{O}_i (\bar{t}) $ (we omit the notation of $ \bar{t} $ in the figure). In the situation of the figure (b), the origin should be on the gray regions including the boundary lines by Corollary 3.4. Thus, the kind of touch like as (b) never occur

Figure 7.  The case of facet-vertex intersection with $ \Lambda_j \subset \mathcal{I}_i (\bar{t}) $. The facet $ \Lambda_{j+1} $ never can be located as the dashed line in (b)

Figure 8.  Location of $ \Lambda_i (\bar{t}) $, $ \Lambda_{i+1} (\bar{t}) $, $ \Lambda_j (\bar{t}) $ and $ \Lambda_{j+1} (\bar{t}) $ under the vertex-vertex intersection between $ \Lambda_i (\bar{t}) $ and $ \Lambda_j (\bar{t}) $. The above figures illustrate the case when (a)$ \varphi_i < \varphi_j < \varphi_i + \pi $, (b)$ \varphi_i + \pi < \varphi_j < \varphi_{i+1} + \pi $, and (c)$ \varphi_{i+1} + \pi < \varphi_j < \varphi_i + 2 \pi $. The gray regions are where $ \Lambda_{j+1} (\bar{t}) $ can be located, and gray shaded regions are where the cross type intersection appears between $ \Lambda_i (\bar{t}) \cup \Lambda_{i+1} (\bar{t}) $ and $ \Lambda_j (\bar{t}) \cup \Lambda_{j+1} (\bar{t}) $ although $ \Lambda_{j+1} (\bar{t}) $ seems to be located from the angle condition

Figure 9.  The case of facet-facet intersection with $ \Lambda_i (\bar{t}) \not\subset \Lambda_j (\bar{t}) $ and $ \Lambda_i (\bar{t}) \not\supset \Lambda_j (\bar{t}) $

Figure 10.  Evolution of a polygonal spiral with a triangle Wulff shape. The figures illustrate the shape of a spiral at $ t = 0.0, 0.1, 0.5, 1.0 $ from top left to bottom right

Figure 11.  Comparison of the profiles of spirals at $ t = 1 $ with respect to different anisotropic mobilities under the same $ \mathcal{W}_\gamma $

Figure 12.  Comparison of profiles between our ODE system(top) and level set method(bottom) for a square spiral at $ t = 0 $(left), $ t = 1 $(center) and $ t = 2 $(right)

Figure 13.  Comparison of profiles between our ODE system(top) and level set method(bottom) for a diagonal spiral at $ t = 0 $(left), $ t = 1 $(center) and $ t = 2 $(right)

Figure 14.  Examination of the orientation of $ L_{k+1} $ with hexagonal $ \mathcal{W}_\gamma $ when $ d_k (T_{k+1}) = \ell_{k}/U $ and $ d_{k-1} (T_{k+1}) < \ell_{k-1} / U $

Figure 15.  The situation when (39) does not hold in Case (i)

Figure 16.  The situation case(a) with $ \varphi_{j+1} \in (\varphi_{i+1} , \varphi_i + \pi) $