Higher-dimensional and dynamic extensions of k-visibility cell decompositions

Investigate extending the k-visibility cell-decomposition paradigm from planar polygons to three-dimensional polyhedral environments and to dynamic settings in which polygon vertices are allowed to move.

Background

The paper develops a cell-decomposition framework for answering k-visibility queries in planar polygons, including polygons with holes. Its decomposition relies on primary vertex horizon lines and secondary hinge transition lines to preserve the combinatorial structure of the k-visibility polygon within each cell. The authors identify extending this framework beyond static planar polygons as an unresolved direction.

Two specific extensions are stated: adapting the cell-decomposition paradigm to three-dimensional polyhedral environments, and handling dynamic polygons whose vertices move. These directions would require determining how visibility-event boundaries, combinatorial mutations, spatial arrangements, and query data structures generalize under higher-dimensional geometry or continuous geometric change.