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* + <sup>Δ</sup>*<sup>t</sup>* <sup>2</sup> . Morphing of the polygon *Pol<sup>t</sup>* into *Polt*+Δ*<sup>t</sup>* 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* + <sup>Δ</sup>*<sup>t</sup>* <sup>2</sup> is illustrated in Figure 5.

**Figure 4.** Additional points facilitating one-to-one transformation (marked by red color).

**Figure 5.** Visualization of one morphing intermediate step at the time moment *t* + *<sup>t</sup>*<sup>0</sup> <sup>2</sup> accordingly to decomposition of boundaries into four parts (marked by different colors).

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 Δ*t* are available. Although there is a correlation of identification numbers between sets of polygons reproducing current and forecasted situations, there is no information in the input data in a merging or splitting case, with part of a pre-image corresponding to a transformed polygon and vice versa. There is no way to exactly retrace the decomposition process for available input data.

In the case of an interpolation approach where the forecasted decomposed polygons are mapped to some parts of the current polygon, the free corridors appearing in the visualization may not reflect the real-life situation and can deviate from it significantly. Therefore, from a safety point of view, aircraft should not be directed through these corridors. Additionally, the time periods between the forecasts are relative short (up to ten minutes). Based on these facts, the mapping of polygons without decomposition of the current polygon in the splitting case and without decomposition of the forecasted polygon in the merging case can be taken as a reasonable approach for transformation. This approach is explained and illustrated in the next subsection.
