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It accelerates boundary and neighbor detection, Delaunay triangulation, and supports adaptive refinement, with implementations in OpenGL for real-world applications including motion planning, UI selection, and dynamic mosaics.",{"@graph":14,"@context":72},[15,34,55],{"@type":16,"itemListElement":17},"BreadcrumbList",[18,23,27,31],{"item":19,"name":20,"@type":21,"position":22},"https://docshare.wps.com","Home","ListItem",1,{"item":24,"name":25,"@type":21,"position":26},"https://docshare.wps.com/document/","Document",2,{"item":28,"name":29,"@type":21,"position":30},"https://docshare.wps.com/document/research-report/","Research & Report",3,{"item":32,"name":10,"@type":21,"position":33},"https://docshare.wps.com/document/fast-computation-of-generalized-voronoi-diagrams-using-graphics-hardware/657884/",4,{"url":32,"name":10,"@type":35,"image":36,"author":41,"headline":10,"publisher":44,"fileFormat":47,"inLanguage":8,"description":12,"dateModified":48,"datePublished":49,"encodingFormat":47,"isAccessibleForFree":50,"interactionStatistic":51},"DigitalDocument",{"url":37,"@type":38,"width":39,"height":40},"https://docshare.wps.com/thumbnails/fast-computation-of-generalized-voronoi-diagrams-using-graphics-hardware/657884.png","ImageObject",300,407,{"name":42,"@type":43},"นรินทร์","Person",{"url":19,"name":45,"@type":46},"DocShare","Organization","application/pdf","2026-10-29","2026-10-26",true,{"@type":52,"interactionType":53,"userInteractionCount":33},"InteractionCounter",{"@type":54},"ViewAction",{"@type":56,"mainEntity":57},"FAQPage",[58,64,68],{"name":59,"@type":60,"acceptedAnswer":61},"How does the method compute generalized Voronoi diagrams using graphics hardware?","Question",{"text":62,"@type":63},"It renders a distance mesh for each Voronoi site and approximates the distance function via polygon rasterization hardware. For each grid sample point, it selects the closest site using polygon scan-conversion and Z-buffer depth comparisons.","Answer",{"name":65,"@type":60,"acceptedAnswer":66},"Which types of Voronoi sites are supported in 2D and 3D?",{"text":67,"@type":63},"The approach constructs distance meshes for points, line segments, polygons, polyhedra, curves, and curved surfaces in both 2D and 3D.",{"name":69,"@type":60,"acceptedAnswer":70},"What capabilities and applications are demonstrated in the paper?",{"text":71,"@type":63},"It supports weighted and farthest-site generalized Voronoi diagrams, efficient computation of Voronoi boundaries and neighbors, and Delaunay triangulation. 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Hoff III, Tim Culver, John Keyser, Ming Lin, Dinesh Manocha  \nUniversity of North Carolina at Chapel Hill  \nDepartment of Computer Science  \nAbstract: We present a new approach for computing generalized 2D and 3D Voronoi diagrams using interpolation-based polygon rasterization hardware. We compute a discrete Voronoi diagram by rendering a three dimensional distance mesh for each Voronoi site. The polygonal mesh is a bounded-error approximation of a (possibly) non-linear function of the distance between a site and a 2D planar grid of sample points. For each sample point, we compute the closest site and the distance to that site using polygon scan-conversion and the Z-buffer depth comparison. We construct distance meshes for points, line segments, polygons, polyhedra, curves, and curved surfaces in 2D and 3D. We generalize to weighted and farthest-site Voronoi diagrams, and present efficient techniques for computing the Voronoi boundaries, Voronoi neighbors, and the Delaunay triangulation of points. We also show how to adaptively refine the solution through a simple windowing operation. The algorithm has been implemented on SGI workstations and PCs using OpenGL, and applied to complex datasets. We demonstrate the application of our algorithm to fast motion planning in static and dynamic environments, selection in complex user-interfaces, and creation of dynamic mosaic effects.  \nCR Categories: I.3.5 [Computer Graphics]: Computational Geometry and Object Modeling; I.3.3 [Computer Graphics]: Picture/Image Generation.  \nAdditional Key Words: Voronoi diagrams, graphics hardware, polygon rasterization, interpolation, motion planning, proximity query, medial axis, OpenGL, framebuffer techniques.  \n1 INTRODUCTION  \nGiven a set of primitives, called Voronoi sites, a Voronoi diagram partitions space into regions, where each region consists of all points that are closer to one site than to any other. Voronoi diagrams have been used in a number of applications including visualization of medical datasets, proximity queries, spatial data manipulation, shape analysis, computer animation, robot motion planning, modeling spatial structures and processes, pattern recognition, and locational optimization. The concept of Voronoi diagrams has been around for at least four centuries, and since the  \ne-mail: {hoff,culver, keyser,lin,[dm}@cs.unc.edu](dm}@cs.unc.edu)[ ](dm}@cs.unc.edu)[WWW:](WWW:) [http://www.cs.unc.edu/](http://www.cs.unc.edu/)~geom/voronoi/  \nCover Plate: Discrete approximation of the generalized Voronoi diagram of four points, a line, a triangle, and one cubic Bézier curve computed interactively on a PC.  \n1970s, algorithms for computing Voronoi diagrams of geometric primitives have been developed in computational geometry and related areas.  \nGood theoretical and practical algorithms are known for computing ordinary Voronoi diagrams of points in any dimension. Ordinary Voronoi diagrams can be generalized in many different ways by using different distance functions and site shapes. A common generalization is to compute the diagram for higher-order sites, such as lines and curves. This greatly increases the complexity since the boundaries of the diagram are composed of high-degree algebraic curves and surfaces, and their intersections; the boundaries of an ordinary point Voronoi diagram are linear. No practically efficient and numerically robust algorithms are known for constructing a topologically consistent, continuous representation of generalized Voronoi diagrams.  \nGiven the practical complexity of computing an exact generalized Voronoi diagram, many authors have proposed approximate algorithms. Interesting approaches include computing the Voronoi diagram of a point-sampling of the sites, adaptively subdividing space to locate the Voronoi boundary, and point-sampling the space to form a volumetric representation of the diagram. In practice, the","cbCaiioOVOX3oyYC","https://ap.wps.com/l/cbCaiioOVOX3oyYC","pdf",3006296,"English","# Abstract\n## 1 Introduction\n## Main Contributions\n## Applications and Implementation\n## Cover Plate","[{\"question\":\"How does the method compute generalized Voronoi diagrams using graphics hardware?\",\"answer\":\"It renders a distance mesh for each Voronoi site and approximates the distance function via polygon rasterization hardware. For each grid sample point, it selects the closest site using polygon scan-conversion and Z-buffer depth comparisons.\"},{\"question\":\"Which types of Voronoi sites are supported in 2D and 3D?\",\"answer\":\"The approach constructs distance meshes for points, line segments, polygons, polyhedra, curves, and curved surfaces in both 2D and 3D.\"},{\"question\":\"What capabilities and applications are demonstrated in the paper?\",\"answer\":\"It supports weighted and farthest-site generalized Voronoi diagrams, efficient computation of Voronoi boundaries and neighbors, and Delaunay triangulation. Implemented with OpenGL, it is applied to fast motion planning, selection in complex user interfaces, and dynamic mosaic effects.\"}]","Fast Computation of Generalized Voronoi Diagrams Using Graphics Hardware | PDF",1792993347,25]