Visibility problems for polyhedral terrains
Journal of Symbolic Computation
The union of moving polygonal pseudodiscs - combinatorial bounds and applications
Computational Geometry: Theory and Applications
ACM Transactions on Graphics (TOG)
The Expected Number of 3D Visibility Events Is Linear
SIAM Journal on Computing
The Envelope of Lines Meeting a Fixed Line and Tangent to Two Spheres
Discrete & Computational Geometry
Lines Avoiding Unit Balls in Three Dimensions
Discrete & Computational Geometry
Transversals to Line Segments in Three-Dimensional Space
Discrete & Computational Geometry
Lines Tangent to Four Triangles in Three-Dimensional Space
Discrete & Computational Geometry
On incremental rendering of silhouette maps of a polyhedral scene
Computational Geometry: Theory and Applications
Lines and Free Line Segments Tangent to Arbitrary Three-Dimensional Convex Polyhedra
SIAM Journal on Computing
Line transversals of convex polyhedra in R3
SODA '09 Proceedings of the twentieth Annual ACM-SIAM Symposium on Discrete Algorithms
On the complexity of umbra and penumbra
Computational Geometry: Theory and Applications
Lines avoiding balls in three dimensions revisited
Proceedings of the twenty-sixth annual symposium on Computational geometry
Lines avoiding balls in three dimensions revisited
Proceedings of the twenty-sixth annual symposium on Computational geometry
| Hi-index | 0.00 |
We present two new fundamental lower bounds on the worst-case combinatorial complexity of sets of free lines and sets of maximal free line segments in the presence of balls in three dimensions. We first prove that the set of maximal non-occluded line segments among n disjoint unit balls has complexity ©(n4), which matches the trivial O(n4) upper bound. This improves the trivial ©(n2) bound and also the ©(n3) lower bound for the restricted setting of arbitrary-size balls [Devillers and Ramos, 2001]. This result settles, negatively, the natural conjecture that this set of line segments, or, equivalently, the visibility complex, has smaller worst-case complexity for disjoint fat objects than for skinny triangles. We also prove an ©(n3) lower bound on the complexity of the set of non-occluded lines among n balls of arbitrary radii, improving on the trivial ©(n2) bound. This new bound almost matches the recent O(n3+µ) upper bound [Rubin, 2010].