# Constant-Time Complete Visibility for Robots with Lights: The Asynchronous Case +
## 6 Stage 1-Interior Depletion
### Stage 1.1-Making Corners of \(\mathbf{P}\) the Corners of \(\mathbf{P}^{\prime}\)
#### 6.2.1 Stage 1.2.1: Moving Four Interior Robots in \(\mathbf{P^{\prime}}\) to \(L_{i}\)
Note that we consider only the robots that are colored internal (or colored start in Stage 1.2.1.1). In Stage 1.2.1.2, when \(t_{1},\ldots,t_{4}\) move to \(EL_{i}\), \(t_{2},t_{3}\) move again in lines \(\overrightarrow{c_{i}t_{2}}\) and \(\overrightarrow{c_{i}t_{3}}\) and \(t_{1}\) and \(t_{4}\) move to the points of \(EL_{i}\) that are not on any line \(\overrightarrow{c_{i}r_{k}}\) joining any interior robot \(r_{k}\) (with color internal) with \(c_{i}\) (note that \(r_{k}\) has not yet moved to \(L_{j}\) or \(EL_{j}\) of any corner \(c_{j}\) of \(\mathbf{P}^{\prime}\)). Therefore, all the interior robots (with color internal) see \(c_{i}\) even after \(t_{1},\ldots,t_{4}\) moved to \(EL_{i}\). We now show that all the interior robots of \(\mathbf{P}^{\prime}\) see all \(t_{1},\ldots,t_{4}\) (in addition to \(c_{i}\)) after \(t_{1},\ldots,t_{4}\) are positioned on \(EL_{i}\). We have from Lemma 1 that any point inside \(EA(c_{i})\) is visible to all the interior robots (with color internal) of \(\mathbf{P}^{\prime}\). Moreover, since \(c_{i}\) is already in \(EA(c_{i})\) due to its move in Stage 1.1 and \(EL_{i}\) joins two points in the neighbor edges of \(c_{i}\) in the perimeter of \(\mathbf{P}^{\prime}\), the lines joining interior robots with \(c_{i}\) intersect \(EL_{i}\) exactly at one point, and no two interior robots of \(\mathbf{P}^{\prime}\) are in any line \(\overrightarrow{c_{i}r_{j}}\) of an interior robot \(r_{j}\). Therefore, \(t_{1},\ldots,t_{4}\) are visible to all the interior robots of \(\mathbf{P}^{\prime}\) even after that are positioned on \(EL_{i}\) (and vice versa). Moreover, since \(t_{1},\ldots,t_{4}\) are in \(EL_{i}\), they are colored eligible. 2.2 Stage 1.2.2: Positioning (at least) Four Interior Robots on the Eligible Lines of the Corners of \(\mathbf{P}^{\prime}\)  
After computing the eligible line \(EL_{i}\) for a corner \(c_{i}\) of \(\mathbf{P}^{\prime}\) in Stage 1.2.1, the goal in this stage is to see whether the four robots on \(EL_{i}\) with color eligible can serve as left and right beacons to apply the Beacon-Directed Curve Positioning framework of Section 3 to reposition the remaining interior robots of \(\mathbf{P}^{\prime}\) (with color internal) to the eligible lines in \(\mathcal{O}(1)\) epochs. If those four robots are positioned such that all the interior robots of \(\mathbf{P}^{\prime}\) inside the corner polygon \(CP(c_{i})\) are within the cone area \(QC(c_{i})\) formed by lines \(\overleftarrow{c_{i}t_{2}},\overleftarrow{c_{i}t_{3}}\), then these robots serve as left and right beacons and this stages finishes with \(c_{i}\) changing its color to corner4. Otherwise, (at most) four robots inside \(CP(c_{i})\) move to \(EL_{i}\) in this stage so that two of them serve as left beacons and two of them serve as right beacons for the Beacon-Directed Curve Positioning framework. Figure 12 illustrates these ideas. The details are as follows. Let \(r_{i}\) be the first robot with color internal to be activated after all the corners of \(\mathbf{P}^{\prime}\) are colored corner3, corner5, or corner with at least a corner colored corner3. Let \(r_{i}\) be closest to corner \(c_{i}\) of \(\mathbf{P}^{\prime}\) than any other corner of \(\mathbf{P}^{\prime}\). Robot \(r_{i}\) sees \(c_{i}\) (Lemma 16) which has color corner3. Robot \(r_{i}\) also sees both the neighboring corners \(c_{i-1},c_{i+1}\) of \(c_{i}\) in \(\mathbf{P}^{\prime}\) and eligible colored robots \(t_{1},t_{2},t_{3},t_{4}\) that are on the eligible line \(EL_{i}\) of \(c_{i}\) (Lemma 16). Let \(QC^{\prime}(r_{i})\) be the cone area of \(\mathbf{P}^{\prime}\) formed by line \(\overleftarrow{c_{i}r_{i}}\) and side \(\overrightarrow{c_{i}c_{i-1}}\) of \(\mathbf{P}^{\prime}\); the left of Figure 12 shows the cone areas \(QC^{\prime}(r_{3})\) and \(QC^{\prime}(r_{4})\) of two internal robots \(r_{3}\) and \(r_{4}\).