Voronoi diagram for multiply-connected polygonal domains 1: algorithm
IBM Journal of Research and Development
Voronoi diagrams—a survey of a fundamental geometric data structure
ACM Computing Surveys (CSUR)
Spatial tessellations: concepts and applications of Voronoi diagrams
Spatial tessellations: concepts and applications of Voronoi diagrams
The Euclidean distance transform
The Euclidean distance transform
The bisector of a point and a plane parametric curve
Computer Aided Geometric Design
The Voronoi diagram of curved objects
Proceedings of the eleventh annual symposium on Computational geometry
New algorithm for medial axis transform of plane domain
Graphical Models and Image Processing
Journal of Computational and Applied Mathematics - Special issue on computational methods in computer graphics
Fast computation of generalized Voronoi diagrams using graphics hardware
Proceedings of the 26th annual conference on Computer graphics and interactive techniques
Geometric constraint solver using multivariate rational spline functions
Proceedings of the sixth ACM symposium on Solid modeling and applications
Voronoi Diagrams of Set-Theoretic solid Models
IEEE Computer Graphics and Applications
Voronoi Diagrams for Planar Shapes
IEEE Computer Graphics and Applications
Problem Reduction to Parameter Space
Proceedings of the 9th IMA Conference on the Mathematics of Surfaces
Building Voronoi Diagrams for Convex Polygons in Linear Expected Time
Building Voronoi Diagrams for Convex Polygons in Linear Expected Time
Design of sculptured surfaces using the b-spline representation
Design of sculptured surfaces using the b-spline representation
Precise Voronoi cell extraction of free-form rational planar closed curves
Proceedings of the 2005 ACM symposium on Solid and physical modeling
The convex hull of freeform surfaces
Computing - Geometric modelling dagstuhl 2002
The predicates of the Apollonius diagram: algorithmic analysis and implementation
Computational Geometry: Theory and Applications - Special issue on robust geometric algorithms and their implementations
Intersecting a freeform surface with a general swept surface
Computer-Aided Design
Computing surface offsets and bisectors using a sampled constraint solver
Proceedings of Graphics Interface 2009
2009 SIAM/ACM Joint Conference on Geometric and Physical Modeling
Medial axis of a planar region by offset self-intersections
Computer-Aided Design
Technical note: Voronoi diagrams of algebraic distance fields
Computer-Aided Design
Computer Aided Geometric Design
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We present robust and efficient algorithms for computing Voronoi diagrams of planar freeform curves. Boundaries of the Voronoi diagram consist of portions of the bisector curves between pairs of planar curves. Our scheme is based on computing critical structures of the Voronoi diagrams, such as self-intersections and junction points of bisector curves. Since the geometric objects we consider in this paper are represented as freeform NURBS curves, we were able to reformulate the solution to the problem of computing those critical structures into the zero-set solutions of a system of nonlinear piecewise rational equations in parameter space. We present a new algorithm for computing error-bounded bisector curves using a distance surface constructed from error-bounded offset approximations of planar curves. This error-bounded algorithm is fast and produces bisector curves that are correct both in topology and geometry. Once bisectors are computed, both local and global self-intersections of the bisector curves are located and trimmed away by solving a system of three piecewise rational equations in three variables. Further, our method computes junction points at which three or more trimmed bisector curves intersect by transforming them into the solutions to a system of piecewise rational equations in the merged parameter space of the planar curves. The bisectors are trimmed at those self-intersection and global junction points. The Voronoi diagram is then computed from the trimmed bisectors using a pruning algorithm. We demonstrate the effectiveness of our approach with several experimental results.