 Info
Exact Voronoi Diagram of Arbitrary Lines in Space

Voronoi Diagram of 4 lines, obtained by rotating one line around the zaxis 
This is a joint project with Geometrica, INRIA Sophia Antipolis  Mediterrane.
Abstract
We introduce a new, efficient, and complete algorithm, and its exact implementation, to compute the Voronoi diagram of lines in threedimensions. This is a major milestone towards the robust construction of the Voronoi diagram of polyhedra. Following the exact geometric computation paradigm, it is guaranteed that we always computes the mathematically correct result. The algorithm is complete in the sense that it can handle all configurations, in particular all degenerate ones. The algorithm requires time and space, where n is the number of lines. The Voronoi diagram is represented by a data structure that permits answering point location queries in expected time. The implementation employs the CGAL packages for constructing arrangements and lower envelopes on parametric surfaces together with advanced algebraic tools.
Bisectors
The bisector of two lines in space is:
(a) 2 generic lines  Hyperbolic paraboloid
(b) 2 parallel lines  a plane
(c) 2 intersecting lines  pair of orthogonal planes
More Diagrams
(a) 4 lines intersecting at a single point
(b) 4 lines  2 intersect and 2 others are parallel to each of them
(c) 5 parallel lines
Open Question and Further Work
 A tight bound on the combinatorial complexity of the Voronoi diagram of lines or line segments in is unknown; it is conjectured that the complexity is nearquadratic; the known lower bound is , but the best known upper bound is . In the case of lines with a fixed number of orientations the upper bound was improved to . Thus, the open question is to close this gap.
 The exact computation is currently based on pure algebraic methods, which slows down the algorithm. Thus, it is necessary to introduce filters that avoid the use of exact arithmetic. Moreover, it would be useful to decrease the complexity of algebraic methods.
 So far it is only possible to compute the Voronoi
diagram of lines. The next steps are to allow also line segments,
points, triangles, and finally polyhedra.
Links
 Michael Hemmer, Ophir Setter, and Dan Halperin.
Constructing the Exact Voronoi Diagram of Arbitrary Lines in Space with Fast PointLocation [preliminary version]
[WWW]
[doi:10.1007/9783642157752_34]
 The package for constructing 3D Voronoi diagram of lines including the point location structure can be download here (as a CGAL package).
 The internal CGAL on which the package is based on can be downloaded here.
 CGAL homepage: www.cgal.org
Contacts
Michael Hemmer



Ophir Setter



Dan Halperin 










