A Method for Topologically Consistent Polygon-to-Point Smooth Transitions in Vario-Scale Modelling

Continuous vario-scale representations provide an effective framework for modelling geographic objects across multiple Levels of Detail (LoDs). In polygon-to-point transitions, straight-skeleton-based generalization may produce spatially disconnected polygon fragments when the resulting LoD Transition Space (LTS) is sliced at intermediate LoDs, leading to topological inconsistencies. This paper proposes a geometry-refinement method based on longest-path LoD normalization. The straight-skeleton network is transformed into a weighted tree in which transition vertices form the nodes and horizontal Euclidean distances define the edge weights. The longest root-to-leaf path is then used as a reference for reassigning the LoD coordinates of the transition vertices. The method is evaluated on three synthetic geometries, a real-world park polygon, and 50 building footprints extracted from OpenStreetMap in Stalowa Wola, Poland. For all evaluated post-normalization intermediate states, no disconnected polygon was observed. In the 50-building experiment, the connectivity rate increased from 93.6% to 100%, while execution time and memory consumption increased by 15.3% and 2.8%, respectively. Re-triangulation increased the numbers of edges and faces by 53.9% and 135.2%. These results provide empirical evidence that longest-path-based LoD normalization can substantially improve topological consistency, although a general connectivity guarantee and the geometric effects of LoD modification require further investigation. Although the pre-normalization connectivity rate varies with input geometry and LoD sampling, all evaluated post-normalization intermediate states remained connected.