# Sparse Arrangements and the Number of Views of Polyhedral Scenes

@article{Berg1997SparseAA, title={Sparse Arrangements and the Number of Views of Polyhedral Scenes}, author={Mark de Berg and Dan Halperin and Mark H. Overmars and Marc J. van Kreveld}, journal={Int. J. Comput. Geom. Appl.}, year={1997}, volume={7}, pages={175-195} }

In this paper we study several instances of the problem of determining the maximum number of topologically distinct two-dimensional images that three-dimensional scenes can induce. To bound this number, we investigate arrangements of curves and of surfaces that have a certain sparseness property. Given a collection of n algebraic surface patches of constant maximum degree in 3-space with the property that any vertical line stabs at most k of them, we show that the maximum combinatorial…

## 32 Citations

### On the Number of Views of Polyhedral Scenes

- Mathematics, Computer ScienceJCDCG
- 2000

It is known that a scene consiing of k convex polyhedra of total complexity n has at most O(n4 k2) distinct orthographic views, and that the number of such views is ?((nk2 + n2)2) in the worst case.…

### Repetitive Hidden Surface Removal for Polyhedra

- Computer ScienceJ. Algorithms
- 1996

Using an off-line data structure of sizemwithn1+??m?n2+?, it is possible to answer on-line hidden-surface-removal queries in timeO(klogn+min{nlogn,kn1+?/m12}), when the scene is composed ofc-oriented polyhedra, which allows dynamic insertion and deletion of polyhedral objects.

### Lines and Free Line Segments Tangent to Arbitrary Three-Dimensional Convex Polyhedra

- Mathematics, Computer ScienceSIAM J. Comput.
- 2007

It is proved that the set of lines tangent to four possibly intersecting convex polyhedra in $\mathbb{R}^3$ with a total of $n$ edges consists of $\Theta(n^2)$ connected components in the worst case.

### On the number of views of translates of a cube and related problems

- Mathematics, Computer ScienceComput. Geom.
- 2004

### Notes on the complexity of exact view graph algorithms for piecewise smooth Algebraic Surfaces

- MathematicsDiscret. Comput. Geom.
- 1998

The view graph of a surface N in 3-space is a graph embedded in the space ν of centers or directions of projection, whose nodes correspond to maximal connected regions of ν which yield equivalent views of N, and it is shown that the exact view graphs of such surfaces can be determined in O(nK(2dimν+1)).

### Region Intervisibility in Terrains

- MathematicsInt. J. Comput. Geom. Appl.
- 2007

This paper presents an algorithm that determines, for any constant ∊ > 0, within O(n1+∊m) time and storage whether or not R1 and R2 are completely intervisible, and gives an O(m3n4) time algorithm to determine whether every point in R1 sees at least one points in R2.

### On the Number of Maximal Free Line Segments Tangent to Arbitrary Three-dimensional Convex Polyhedra

- Mathematics
- 2005

We prove that the lines tangent to four possibly intersecting convex polyhedra in $ ^3$ with $n$ edges in total form $\Theta(n^2)$ connected components in the worst case. In the generic case, each…

### On incremental rendering of silhouette maps of polyhedral scene

- Mathematics, Computer ScienceSODA '00
- 2000

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