Generalized Voronoi Diagrams and Lie Sphere Geometry
John Edwards, Tracy Payne, and Elena Schafer

TL;DR
This paper employs Lie sphere geometry to unify and extend the understanding of various generalized Voronoi diagrams, including classical, power, Apollonius, and medial axes, through convex hull and algebraic approaches.
Contribution
It introduces a Lie geometric framework that generalizes and unifies multiple types of Voronoi diagrams, linking them to convex hull problems and algebraic structures.
Findings
Unified geometric description of generalized Voronoi diagrams
Algorithms for computing these diagrams and their complexity analysis
Extension of classical diagrams to include spheres and half-spaces
Abstract
We use Lie sphere geometry to describe two large categories of generalized Voronoi diagrams that can be encoded in terms of the Lie quadric, the Lie inner product, and polyhedra. The first class consists of diagrams defined in terms of extremal spheres in the space of Lie spheres, and the second class includes minimization diagrams for functions that can be expressed in terms of affine functions on a higher-dimensional space. These results unify and generalize previous descriptions of generalized Voronoi diagrams as convex hull problems. Special cases include classical Voronoi diagrams, power diagrams, order and farthest point diagrams, Apollonius diagrams, medial axes, and generalized Voronoi diagrams whose sites are combinations of points, spheres and half-spaces. We describe the application of these results to algorithms for computing generalized Voronoi diagrams and find the…
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Taxonomy
TopicsAdvanced Topics in Algebra · Mathematics and Applications · Advanced Differential Geometry Research
