# Innovative Integration of Severe Weather Forecasts into an Extended Arrival Manager
## 3 AMAN Air Traffic Controller Support
### Dynamic Severe Weather Visualization on a Traffic Display
#### 3.4.2 Moving and Morphing Algorithm Description  
The first mentioned transformation of one polygon into another provides the basis for the two remaining types of transformations, as well as the subsequent morphing of the set of polygons at time \(t\) into the set of polygons at time \(t+\Delta t\) over the time period \(\Delta t\). Therefore, this should be primarily considered. Decomposition of Polygon Boundaries  
To morph one polygon into another, the boundaries of considered polygons are decomposed into four corresponding parts to obtain a more natural interpolation of the transformation that avoids significant turns and rotations of the boundary during interpolation. For this purpose, four sets of vertices with minimal and maximal abscissas and with minimal and maximal ordinates have to be found for the current and for the forecasted polygon. These sets can contain more than one point, and some sets can coincide. If the considered set consists of more than one point, the simplest solution is to take the first point from each set of vertices in the vertex index increase direction. Another option is to choose a point from this set, which provides balanced length for the boundary parts. Boundary parts obtained in this manner are illustrated in Figure 3. Here, \(Pol^{t}\) and \(Pol^{t+\Delta t}\) are polygons at time moments \(t\) and \(t+\Delta t\), respectively. The points \(E^{t}_{j}\), \(i=1,\ldots,8\) and \(E^{t+\Delta t}_{j}\), \(j=1,\ldots,9\) are the vertices and \(E^{*_{Top}}\), \(E^{*Right}\), \(E^{*Bottom}\), \(E^{*Left}\) are subsets of vertices with minimal or maximal first or second coordinates of the polygons \(Pol^{t}\) and \(Pol^{t+\Delta t}\). After boundary decomposition, morphing of the boundary parts is performed through the linear interpolation of the vertices in \(s\) discrete time steps of the duration \(\frac{\Delta t}{s}\). To perform the interpolation, the boundary parts are balanced in order to establish one-to  
Figure 3: Decomposition of boundaries into four parts (marked by different colors) for the actual polygon \(Pol^{t}\) and the next (forecasted) one \(Pol^{t\ +\Delta t}\). one correspondence between them. This means that the boundary part with the smallest number of vertices receives additional points uniformly distributed along the edges so that, on each edge, the quotient number of additional points plus one additional point on the remaining number of edges in the vertex index increase direction is selected. The described approach is illustrated in Figure 4. Here, the piecewise linear curve \(A_{1}A_{3}\) is transformed into the piecewise linear curve \(B_{1}B_{10}\). The quotient of the division of ten by three is equal to three and the remainder is equal to one. Therefore, the edge \(A_{1}A_{2}\) gets three plus one additional points, and three additional equidistant points are selected on the \(A_{2}A_{3}\) edge. They are marked red in Figure 4.  
Moving and Morphing of a Single Polygon  
The described approach provides an identical number of \"vertices\" on both linear curves to perform one-to-one linear interpolation in \(s\) discrete steps. The middle polyline in Figure 4 illustrates the interpolated curve at the time moment \(t+\frac{\Delta t}{2}\). Morphing of the polygon \(Pol^{t}\) into \(Pol^{t+\Delta t}\) is realized through interpolation of the transformation of four corresponding boundary parts in accordance with the approach visualized in Figure 4. The interpolated polygon at time \(t+\frac{\Delta t}{2}\) is illustrated in Figure 5.  
The input data to perform interpolation represent the current and forecasted locations of cells, i.e., only the current situation and a forecasted result of transformation after time period \(\Delta t\) are available. Although there is a correlation of identif  
Figure 4: Additional points facilitating one-to-one transformation (marked by red color).