The expected number of 3D visibility events is linear - Archive ouverte HAL Access content directly
Journal Articles SIAM Journal on Computing Year : 2003

## The expected number of 3D visibility events is linear

(1) , (2) , (3) , (3) , (3) , (4) , (3)
1
2
3
4
Olivier Devillers
Vida Dujmovic
• Function : Author
Hazel Everett
• Function : Author
• PersonId : 830545
Xavier Goaoc
Sylvain Lazard
Hyeon-Suk Na
• Function : Author

#### Abstract

In this paper, we show that, amongst $n$ uniformly distributed unit balls in $\mathbb{R}^3$, the expected number of maximal non-occluded line segments tangent to four balls is linear. Using our techniques we show a linear bound on the expected size of the visibility complex, a data structure encoding the visibility information of a scene, providing evidence that the storage requirement for this data structure is not necessarily prohibitive. These results significantly improve the best previously known bounds of $O(n^{8/3})$. Our results generalize in various directions. We show that the linear bound on the expected number of maximal non-occluded line segments that are not too close to the boundary of the scene and tangent to four unit balls extends to balls of various but bounded radii, to polyhedra of bounded aspect ratio, and even to non-fat 3D objects such as polygons of bounded aspect ratio. We also prove that our results extend to other distributions such as the Poisson distribution. Finally, we indicate how our probabilistic analysis provides new insight on the expected size of other global visibility data structures, notably the aspect graph.

#### Domains

Computer Science [cs] Other [cs.OH]

### Dates and versions

inria-00099810 , version 1 (15-12-2009)

### Identifiers

• HAL Id : inria-00099810 , version 1
• DOI :

### Cite

Olivier Devillers, Vida Dujmovic, Hazel Everett, Xavier Goaoc, Sylvain Lazard, et al.. The expected number of 3D visibility events is linear. SIAM Journal on Computing, 2003, 32 (6), pp.1586-1620. ⟨10.1137/S0097539702419662⟩. ⟨inria-00099810⟩

### Export

BibTeX TEI Dublin Core DC Terms EndNote Datacite

151 View