## New Bounds for Lower Envelopes in Three Dimensions, with Applications to Visibility in Terrains (1997)

Venue: | Geom |

Citations: | 60 - 25 self |

### BibTeX

@ARTICLE{Halperin97newbounds,

author = {Dan Halperin and Micha Sharir},

title = {New Bounds for Lower Envelopes in Three Dimensions, with Applications to Visibility in Terrains},

journal = {Geom},

year = {1997},

volume = {12},

pages = {313--326}

}

### Years of Citing Articles

### OpenURL

### Abstract

We consider the problem of bounding the complexity of the lower envelope of n surface patches in 3-space, all algebraic of constant maximum degree, and bounded by algebraic arcs of constant maximum degree, with the additional property that the interiors of any triple of these surfaces intersect in at most two points. We show that the number of vertices on the lower envelope of n such surface patches is O(n 2 \Delta 2 c p log n ), for some constant c depending on the shape and degree of the surface patches. We apply this result to obtain an upper bound on the combinatorial complexity of the `lower envelope' of the space of all rays in 3-space that lie above a given polyhedral terrain K with n edges. This envelope consists of all rays that touch the terrain (but otherwise lie above it). We show that the combinatorial complexity of this ray-envelope is O(n 3 \Delta 2 c p log n ) for some constant c; in particular, there are at most that many rays that pass above th...

### Citations

423 | Davenport-Schinzel Sequences and Their Geometric Applications - Sharir, Agarwal - 1995 |

403 | Applications of random sampling in computational geometry ii
- Clarkson
- 1988
(Show Context)
Citation Context ... of these surfaces is O(n 2 \Delta 2 c p log n ), for some constant c that depends on the degree and shape of the given surfaces. The proof is not difficult, and relies on the randomized technique of =-=[7, 16]-=- for obtaining generalized `( k)-set' bounds in arrangements. This result still leaves a small gap from the conjectured complexity, but is nevertheless a significant and rather decisive step towards t... |

144 |
Combinatorial complexity bounds for arrangements of curves and spheres, Discrete Comput
- Clarkson, Edelsbrunner, et al.
- 1990
(Show Context)
Citation Context ...ma. We claim that the total number of such vertices that lie on the boundary of the fixed oe is O( q+2 (n)), for some constant q depending on the degree and shape of the surfaces. This is shown as in =-=[5, 6]-=-: Let ff be one of the (constant number of) algebraic arcs that form the boundary of oe, and let H be the vertical surface formed by the union of all vertical rays whose bottom endpoints lie on ff. Fo... |

107 | Sharir M., Nonlinearity of Davenport-Schinzel sequences and of generalized path compression schemes, Combinatorica 6
- Hart
- 1986
(Show Context)
Citation Context ...unctions intersect in at most s points, then the complexity of their envelope is at mostss+2 (n), which is the maximum length of Davenport-Schinzel sequences of order s + 2 composed of n symbols (see =-=[2, 11]-=- for more details), and is only slightly super-linear in n for any fixed s. The conjecture in 3 dimensions attempts to extend this bound, and asserts that the complexity of the envelope is O(n q (n)),... |

83 |
Almost tight upper bounds for lower envelopes in higher dimensions, Discrete Comput
- Sharir
- 1994
(Show Context)
Citation Context ...nts. This result still leaves a small gap from the conjectured complexity, but is nevertheless a significant and rather decisive step towards the establishment of the conjecture. In a companion paper =-=[17]-=- the technique presented in this paper is extended to obtain similar, almost-tight bounds in more general setups, both when the maximum number of points of intersection between a triple of surfaces ca... |

80 |
Sharp upper and lower bounds for the length of general Davenport-Schinzel sequences
- Agarwal, Sharir, et al.
- 1989
(Show Context)
Citation Context ...unctions intersect in at most s points, then the complexity of their envelope is at mostss+2 (n), which is the maximum length of Davenport-Schinzel sequences of order s + 2 composed of n symbols (see =-=[2, 11]-=- for more details), and is only slightly super-linear in n for any fixed s. The conjecture in 3 dimensions attempts to extend this bound, and asserts that the complexity of the envelope is O(n q (n)),... |

75 |
Visibility problems for polyhedral terrains
- Cole, Sharir
- 1989
(Show Context)
Citation Context ...ple-contact views, and observe that the number of views that we seek to bound is proportional to the complexity of the arrangement of these arcs within S 2 . We next apply a result of Cole and Sharir =-=[8]-=-, which, rephrased in the context under discussion, states that each meridian of S 2 crosses at most k = O(n 4 (n)) arcs. As shown in [4], this implies that the complexity of the arrangement of these ... |

61 |
On k-sets in arrangements of curves and surfaces
- Sharir
- 1991
(Show Context)
Citation Context ... of these surfaces is O(n 2 \Delta 2 c p log n ), for some constant c that depends on the degree and shape of the given surfaces. The proof is not difficult, and relies on the randomized technique of =-=[7, 16]-=- for obtaining generalized `( k)-set' bounds in arrangements. This result still leaves a small gap from the conjectured complexity, but is nevertheless a significant and rather decisive step towards t... |

31 | A singly exponential stratification scheme for real semialgebraic varieties and its applications - Chazelle, Edelsbrunner, et al. - 1989 |

29 | Kreveld, Sparse arrangements and the number of views of polyhedral scenes
- Berg, Halperin, et al.
- 1991
(Show Context)
Citation Context ...\Delta 2 c p log n ) for some constant c; in particular, there are at most that many rays that pass above the terrain and touch it in 4 edges. This bound, combined with the analysis of de Berg et al. =-=[4]-=-, gives an upper bound (which is almost tight in the worst case) on the number of topologically-different orthographic views of such a terrain. Work on this paper by the first author has been supporte... |

24 |
The upper envelope of piecewise linear functions and the boundary of a region enclosed by convex plates: combinatorial analysis, Discrete Comput
- Pach, Sharir
- 1989
(Show Context)
Citation Context ...the given patches. The conjecture appears to be extremely difficult, and has been proven for families of only a few types of surfaces or surface patches, such as triangles, and a few other types (see =-=[13, 15]-=-). Better bounds are known for the special cases of planes and balls. The problem in general has been wide open; in fact, no general bounds better than O(n 3 ) were known so far. In this paper we obta... |

20 |
personal communication
- Sridhar, Klein
(Show Context)
Citation Context ... readily obtain the bound asserted in the theorem. 2 Remarks: (1) The bound of Theorem 3.1 has been independently obtained by Pellegrini [14], using a different proof technique. (2) Recently, de Berg =-=[3]-=- has shown a lower bound construction of complexity \Omega\Gamma n 3 ) for the envelope of LK , implying that our upper bound is almost tight in the worst case. The construction consists of an almost ... |

18 |
Algorithmic Motion Planning via Arrangements of Curves and Surfaces
- Halperin
(Show Context)
Citation Context ...f the motion planning problem for an arbitrary polygonal object moving (translating and rotating) in a 2-D polygonal environment, leading to near-quadratic bounds on that complexity. The dissertation =-=[9]-=- has studied several special cases of this motion planning problem, and obtained better, often near-quadratic bounds in these cases, but no subcubic bounds were known for the general problem, as just ... |

17 | On the number of views of polyhedral terrains
- Agarwal, Sharir
- 1994
(Show Context)
Citation Context ...ase of EK . Nevertheless, Agarwal and Sharir have recently obtained an almost tight bound on the number of topologically-different perspective views of a polyhedral terrain using a different approach =-=[1]-=-. 4 Conclusion The new bounds obtained in Section 2 for the complexity of the lower envelope of surface patches in 3-space push the frontier of research on these problems a step further. In some intui... |

14 |
Schubert calculus
- Kleiman, Laksov
- 1972
(Show Context)
Citation Context ...o a line that passes through the four edges e 0 , e 1 , e 2 and e 3 , and it is well known that there can be at most two such lines, assuming that these four edges are in general position (see, e.g., =-=[12]-=-). Condition (iv) can be enforced by assuming the terrain K to be in general position. We argue, as in [17], that the maximum complexity of the envelope is achieved, up to a constant factor, when the ... |

12 |
On the two-dimensional Davenport-Schinzel problem
- Schwartz, Sharir
- 1996
(Show Context)
Citation Context ...ven surfaces are in general position (see Section 2 for precise definitions). The lower envelope, when projected onto the xy-plane, generates a planar map M, called the minimization diagram of \Sigma =-=[15]-=-, with the property that over each face of M the envelope is attained by a single patch (or by no patch at all), over each edge the envelope is attained by two patches simultaneously or by the boundar... |

8 |
On lines missing polyhedral sets
- Pellegrini
- 1993
(Show Context)
Citation Context ...at the conditions required by our analysis hold in this application. Our bound for the complexity of the space of lines passing above a terrain was recently, and independently, obtained by Pellegrini =-=[14]-=-, but it is not clear whether his result can be extended to the space of rays passing above a terrain. This bound on the complexity of the envelope of the space of rays over a terrain has an interesti... |

6 |
Near-quadratic bounds for the motion planning problem for a polygon in a polygonal environment
- Halperin, Sharir
- 1993
(Show Context)
Citation Context ...ere known for the general problem, as just stated. This extension of our result requires considerably more involved analysis than the one given in this paper, and it is presented in a companion paper =-=[10]-=-. Finally, the new technique developed in this paper will be extended in the companion paper [17] to obtain similar almost tight bounds for the complexity of the envelope in cases where the maximum nu... |