A motivating example

Figure 1: An example of shadows and their indexing/labeling. (a) The set of spotlights and the path to be followed by the darker (orange) colored spotlight. (b) Initially, only shadow $s_2$ and unbounded shadow $s_1$ exist. (c) A new shadow $s_3$ appears. (d) $s_2, s_3$ merge into a single shadow $s_4$ . (e) $s_4$ splits into new shadows $s_5, s_6$ . (f) $s_6$ disappears.

To offer some intuition behind the mathematical definitions soon to arrive, let us look at the example shown in Fig. 1(a). With the intention of guarding a planar region, spotlights are cast on the ground, creating a set of illuminated discs as shown. Assume that only the darker (orange) colored disc of light moves and follows the dashed line. For any position of the moving spotlight, the combined, illuminated set can be thought of as the field-of-view or the visible region. Its complement in the plane is the shadow region, where we cannot directly observe targets. Initially, there are two connected components, labeled $s_1$ (unbounded) and $s_2$ , in the shadow region. As the spotlight moves along the dashed line, we observe that shadow components may appear, disappear, merge, and split, as illustrated in Fig. 1(b) to (f). Fig. 2 is an abstract illustration of the evolution of the shadow components through time. Here $W = \mathbb{R}^2$ is the workspace and $T$ is the time axis. We are now ready to define shadows and component events in a more general and formal manner.

Figure 2: Evolution of shadows: $t_1$ : $s_3$ appears, $t_2$ : $s_2, s_3$ merge into $s_4$ , $t_3$ : $s_4$ splits into $s_5, s_6$ , $t_4$ : $s_6$ disappears.