Sparse Arrangements and the Number of Views of Polyhedral Scenes

  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.},
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… 

On the Number of Views of Polyhedral Scenes

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

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

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.

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

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

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

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

Visibility maps of segments and triangles in 3D

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Arrangements and Their Applications



Visibility Problems for Polyhedral Terrains

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We show that the number of topologically different orthographic views of a polyhedral terrain withn edges isO(n5+ɛ), and that the number of topologically different perspective views of such a terrain

Creating The Perspective Projection Aspect Graph Of Polyhedral Objects

  • J. StewmanK. Bowyer
  • Mathematics
    [1988 Proceedings] Second International Conference on Computer Vision
  • 1988
An algorithm is presented for constructing the perspective projection aspect graph of polyhedra, in which the faces of the object lie, and three types of surfaces are involved: object planes, auxiliary planes, defined by the visual interaction of edge-vertex pairs, and auxiliary quadric surfaces, definedby the visual interacting of edge triplets.

Applications of random sampling in computational geometry, II

Asymptotically tight bounds for (≤k)-sets are given, which are certain halfspace partitions of point sets, and a simple proof of Lee's bounds for high-order Voronoi diagrams is given.

Planar realizations of nonlinear davenport-schinzel sequences by segments

  • Ady Wiernik
  • Mathematics
    27th Annual Symposium on Foundations of Computer Science (sfcs 1986)
  • 1986
A construction of a set G of n segments for which YG consists of Ω(nα(n)) subsegments is presented, proving that the Hart-Sharir bound is tight in the worst case.

Approximations and optimal geometric divide-and-conquer

We give an efficient deterministic algorithm for computing ?-approximations and ?-nets for range spaces of bounded VC-dimension. We assume that an n-point range space ? = (X, R) of VC-dimension d is

An algorithm for constructing the aspect graph

  • W. H. PlantingaC. Dyer
  • Computer Science, Mathematics
    27th Annual Symposium on Foundations of Computer Science (sfcs 1986)
  • 1986
This paper gives upper and lower bounds on the maximum size of aspect graphs and gives worstcase optimal algorithms for their construction, first in the convex case and then in the general case and shows a different way to label the aspect graph.

Combinatorics and Algorithms of Arrangements

In this chapter, the combinatorial structure of arrangements of algebraic curves or surfaces in low-dimensional Euclidean space is studied, taken from the theory of motion planning in robotics.

New bounds for lower envelopes in three dimensions, with applications to visibility in terrains

An upper bound is obtained on the combinatorial complexity of the “lower envelope” of thespace of all rays in 3-space that lie above a given polyhedral terrain with edges with the additional property that the interiors of any triple of these surfaces intersect in at most two points.

Aspect graphs: An introduction and survey of recent results

  • K. BowyerC. Dyer
  • Computer Science
    ISPRS International Conference on Computer Vision and Remote Sensing
  • 1990
A tutorial introduction to the aspect graph is presented, the current state of the art in algorithms for automatically constructing aspect graphs is surveyed, and some possible applications of aspect graphs in computer vision and computer graphics are described.