Geodesic isolines derived from polylines constitute a crucial element within geographic information systems (GIS), playing a pivotal role in enhancing the understanding of geographical terrains. Current methods for delineating isolines sourced from polylines on discrete meshes often rely on simplistic linear interpolation. However, these methods fall short in accuracy due to the complex, non-linear nature of geodesic distance fields, thereby inadequately capturing intricate topological features present in real isolines. To tackle this challenge, we demonstrate that Apollonius diagrams can effectively encode the geometric attributes of isolines on meshes and extract the isolines using the Apollonius diagrams with geodesic metric. Moreover, exact geodesic computation is computationally intensive and memory-demanding. In response, we introduce a graph-based approach enhanced by Steiner point insertion, offering a practical method for computing geodesic distances. Drawing on these strategies, we introduce an accurate and efficient algorithm for polyline-sourced isoline computation on triangle meshes. Comprehensive evaluations indicate that our approach yields significantly more accurate geodesic isolines compared to the commonly employed linear interpolation.

Voronoi diagrams on triangulated surfaces based on the geodesic metric play a key role in many applications of computer graphics. Previous methods of constructing such Voronoi diagrams generally depended on having an exact geodesic metric. However, exact geodesic computation is time-consuming and has high memory usage, limiting wider application of geodesic Voronoi diagrams (GVDs). In order to overcome this issue, instead of using exact methods, we reformulate a graph method based on Steiner point insertion, as an effective way to obtain geodesic distances. Further, since a bisector comprises hyperbolic and line segments, we utilize Apollonius diagrams to encode complicated structures, enabling Voronoi diagrams to encode a medial-axis surface for a dense set of boundary samples. Based on these strategies, we present an approximation algorithm for efficient Voronoi diagram construction on triangulated surfaces. We also suggest a measure for evaluating similarity of our results to the exact GVD. Although our GVD results are constructed using approximate geodesic distances, we can get GVD results similar to exact results by inserting Steiner points on triangle edges. Experimental results on many 3D models indicate the improved speed and memory requirements compared to previous leading methods.