Hierarchical Polygon-to-Point Collapsing for Multi-Scale Representation Based on the Straight Skeleton

This paper presents a LoD transition space for the dimensional collapse of a polygon into point(s) within a structured multi-scale framework. Unlike traditional cartographic generalisation, where topological relationships may be modified to improve map readability, the proposed method follows a model-based representation in which transitions are derived explicitly from straight skeleton events. The methodology uses the straight skeleton to generate a sequence of shrinking stages governed by edge and split events, each of which defines both a topological transformation and its corresponding geometric change. Based on these event-driven transitions, intermediate Levels of Detail (LoDs) are constructed and organized hierarchically. The resulting hierarchy is then mapped to the Dual Half-Edge (DHE) structure, where the primal space represents successive geometric states and the dual space represents their hierarchical relations along the scale dimension. This integration produces a unified 2D+1D representation that supports a continuous transition from polygon to point. In addition to its relevance for vario-scale cartography, the proposed framework has potential applicability in domains requiring structured shape transformation, such as animation and procedural modelling.