# 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 \(L_{i}\)
This is because they see all the corners of \(\mathbf{P^{\prime}}\). Observe that it is possible for some of the corners of \(\mathbf{P^{\prime}}\) to have fewer than four robots (or even no robot) on \(L_{i}\) even after all robots in \(Q_{i}\) have color internal. Those corners change their color directly to corner5 from corner2. Stage 1.2.1.2-Computing Eligible Lines for the Corners of \(\mathbf{P^{\prime}}\): We describe how to compute \(EL_{i}\) for \(c_{i}\). Let \(t_{1},t_{2},t_{3},t_{4}\) be the four robots in \(L_{i}\) of corner \(c_{i}\) (Stage 1.2.1.1) with \(t_{2}\) and \(t_{3}\) between \(t_{1}\) and \(t_{4}\), and \(t_{2},t_{3}\) being closer to \(t_{1},t_{4}\), respectively (Figure 0(a)). We ask \(t_{1}\) and \(t_{4}\) to move to the lines \(\overline{c_{i}t_{2}}\) and \(\overline{c_{i}t_{3}}\), respectively, assuming color transit_moving. Robots \(t_{1},t_{4}\) perform this move only when they have color transit and \(c_{i}\) has color corner22. The position they move to in those lines is the \(1/8\)-th point from \(c_{i}\) to \(t_{2}\) and \(t_{3}\), respectively. They then change their color to transit1 (Figure 0(b)). After \(c_{i}\) sees both \(t_{1},t_{4}\) with color transit1, it computes \(EA(c_{i})\) and a point \(x_{i}\) on \(\overline{c_{i}c_{i-1}}\) (or \(y_{i}\) on \(\overline{c_{i}c_{i+1}}\)) so that the line, say \(L_{i}^{\prime}\), parallel to \(\overline{t_{1}t_{4}}\) passing through \(x_{i}\) (or \(y_{i}\)) crosses \(EA(c_{i})\). According to the construction, \(\overline{t_{1}t_{4}}\) is parallel to \(\overline{t_{2}t_{3}}\), and also parallel to \(\overline{c_{i-1}c_{i+1}}\). Let \(x_{i}\) on \(\overline{c_{i}c_{i-1}}\) be the point so that \(L_{i}^{\prime}\) crosses \(EA(c_{i})\). Observe that \(L_{i}^{\prime}\) is in fact the eligible line \(EL_{i}\). Corner \(c_{i}\) then moves to \(x_{i}\) (the procedure for \(c_{i}\) moving to point \(y_{i}\) is analogous) assuming color corner22_moving (Figure 0(c)) and changes its color to corner23. Let \(op_{i}\) be the position of \(c_{i}\) before it moves to \(x_{i}\). We now describe a technique to put all \(t_{1},t_{2},t_{3},t_{4}\) on \(L_{i}^{\prime}\) (which is \(EL_{i}\)) so that the interior robots of \(\mathbf{P^{\prime}}\) can recognize it as \(EL_{i}\). Let \(t_{1}\) be closer to \(c_{i}\) than \(t_{4}\) from the new position \(x_{i}\) of \(c_{i}\) (the case of \(t_{4}\) being closer to \(c_{i}\) than \(t_{1}\) is analogous). Robot \(t_{1}\) moves to the intersection point of \(L_{i}^{\prime}\) and \(\overline{t_{1}t_{2}}\) assuming color transit1_moving (Figure 0(d)) and then changes its color to transit2 when it becomes active next time. After \(c_{i}\) sees \(t_{1}\) colored transit2, it moves back to its previous position \(op_{i}\) (where it was colored corner22) assuming color corner23_moving (Figure 0(e)). Although \(c_{i}\) has no memory of \(op_{i}\), it can compute \(op_{i}\) since \(op_{i}\) is the intersection point of lines \(\overline{t_{1}t_{2}}\) and \(\overline{t_{4}t_{3}}\). Robot \(c_{i}\) then assumes corner24. After this, \(t_{4}\) with color transit1 moves to the intersection point of \(L_{i}^{\prime}\) and \(\overline{t_{4}t_{3}}\) assuming color transit1_moving (Figure 0(f)). It then assumes color transit2. Let \(op_{1},op_{4}\) be the current positions of \(t_{1},t_{4}\), respectively. The robots \(t_{1}\) and \(t_{4}\) (after taking color transit2) move either left or right in \(L_{i}^{\prime}\) to make room for robots \(t_{2}\) and \(t_{3}\) to move to \(L_{i}^{\prime}\) without blocking any internal colored robots to see \(c_{i}\) and also the robots\(t_{1},t_{2},t_{3},t_{4}\) on \(L^{\prime}_{i}\). Robot \(t_{1}\) (and similarly \(t_{4}\)) moves as follows. Let \(\overleftrightarrow{c_{i}t_{1}}\) be a line that connects \(t_{1}\) with \(c_{i}\).