## Contour Edge Analysis for Polyhedron Projections (1997)

Venue: | GEOMETRIC MODELING: THEORY AND PRACTICE |

Citations: | 20 - 3 self |

### BibTeX

@INPROCEEDINGS{Kettner97contouredge,

author = {Lutz Kettner and Emo Welzl},

title = {Contour Edge Analysis for Polyhedron Projections},

booktitle = {GEOMETRIC MODELING: THEORY AND PRACTICE},

year = {1997},

pages = {379--394},

publisher = {Springer}

}

### Years of Citing Articles

### OpenURL

### Abstract

Given a polyhedron (in 3-space) and a view point, an edge of the polyhedron is called contour edge, if one of the two incident facets is directed towards the view point, and the other incident facet is directed away from the view point. Algorithms on polyhedra can exploit the fact that the number of contour edges is usually much smaller than the overall number of edges. The main goal of this paper is to provide evidence for (and quantify) the claim, that the number of contour edges is small in many situations. An asymptotic analysis of polyhedral approximations of a sphere with Hausdorff distance " shows that while the required number of edges for such an approximation grows like \Theta(1="), the number of contour edges in a random orthogonal projection is \Theta(1= p " ). In an experimental study we investigate a number of polyhedral objects from several application areas. We analyze the expected number of contour edges and the expected number of intersections of contour edges in ...

### Citations

691 | Surface reconstruction from unorganized points
- Hoppe, DeRose, et al.
- 1992
(Show Context)
Citation Context ...se smooth surface has been computed and fandisk3 is a piecewise linear approximation [12]. The coefficient c 1 is 3:5 and 3:0, respectively. bunny is a data set from a 3d-scanner. It has been used in =-=[5, 11]-=-. face in Figure 11 is generated with a marching cube algorithm. Although this algorithm has produced a remarkable band structure in the reconstruction, Figure 10. Examples from mesh generation from C... |

541 | Stuetzle Multiresolution analysis for arbitrary meshes
- Eck, DeRose, et al.
- 1995
(Show Context)
Citation Context ...se smooth surface has been computed and fandisk3 is a piecewise linear approximation [12]. The coefficient c 1 is 3:5 and 3:0, respectively. bunny is a data set from a 3d-scanner. It has been used in =-=[5, 11]-=-. face in Figure 11 is generated with a marching cube algorithm. Although this algorithm has produced a remarkable band structure in the reconstruction, Figure 10. Examples from mesh generation from C... |

369 | Mesh Optimization
- Hoppe
(Show Context)
Citation Context ...es the tolerated error bound in the approximation. The coefficient in c 1 p n is here 21:7 and 30 for cola 0.05 and cola 0.005, respectively. fandisk1 is an optimized mesh from surface reconstruction =-=[13]-=-. From that, a piecewise smooth surface has been computed and fandisk3 is a piecewise linear approximation [12]. The coefficient c 1 is 3:5 and 3:0, respectively. bunny is a data set from a 3d-scanner... |

277 | Piecewise smooth surface reconstruction
- Hoppe, DeRose, et al.
- 1994
(Show Context)
Citation Context ... and cola 0.005, respectively. fandisk1 is an optimized mesh from surface reconstruction [13]. From that, a piecewise smooth surface has been computed and fandisk3 is a piecewise linear approximation =-=[12]-=-. The coefficient c 1 is 3:5 and 3:0, respectively. bunny is a data set from a 3d-scanner. It has been used in [5, 11]. face in Figure 11 is generated with a marching cube algorithm. Although this alg... |

271 | Convex Polytopes
- Grünbaum
- 1967
(Show Context)
Citation Context ...ar on the boundary of the projection. Since these events are disjoint for all vertices, P 4 i=1 2ff v i is the probability that T projects to a triangle. Using the so-called Gram-Sommerville idendity =-=[9, 21]-=- we can rewrite this sum as 3 2 4 X i=1 ff v i = 2 6 X i=1 ff e i \Gamma 2 : Plugging this into (1), we get Pr[e 1 intersects e 2 ] = 1 2 / 2ff e1 + 2ff e2 \Gamma / 2 6 X i=1 ff e i \Gamma 2 !! : Lemm... |

206 | A characterization of ten hidden surface algorithms, Computing Surveys 6
- Sutherland, Sproull, et al.
- 1974
(Show Context)
Citation Context ...ment s. This shadow is an unbounded facet of the shadow volume. They can be used to cast shadows onto other objects or even the object itself. The exact solution provided by an object space algorithm =-=[19]-=- avoids the drawback of aliasing effects of raster algorithms for shadow casting. We use a line sweep algorithm to compute all intersections of the projected contour edges in the projection plane. It ... |

68 | The notion of quantitative invisibility and the machine rendering of solids
- Appel
- 1967
(Show Context)
Citation Context ...ion. Algorithms on polyhedra (or polyhedral scenes) can exploit the fact that the number of contour edges is usually much smaller than the overall number of edges. This was first pointed out by Appel =-=[1]-=- when describing an object space hidden surface removal algorithm. The main goal of this paper is to provide evidence for (and quantify) the claim, that the number of contour edges is small in many si... |

49 | The CGAL kernel: A basis for geometric computation
- Fabri, Giezeman, et al.
- 1996
(Show Context)
Citation Context ...l algorithms [19]. It is intended to provide implementations for the Cgal project, a coordinated effort of seven research groups in Europe for constructing a Computational Geometry Algorithms Library =-=[6, 16, 3]-=-. The approach we are focussing on uses a line sweep algorithm for hidden surface elimination in the projection plane as in the example above for the silhouette computation. A similar line sweep consi... |

49 |
Aspects of approximation of convex bodies
- Gruber
- 1993
(Show Context)
Citation Context ...have n c = P e2E 1 \Gamma 2ff e = O( p " )jEj due to Lemmas 2.2 and 2.3. The number of edges required for an approximation of the unit sphere with Hausdorff distance " is known to be \Theta(=-=1=") (see [8]). We-=- conclude that Theorem 2.5. Let n be the number of edges of an optimal convex "-approximation of the unit sphere and n c the expected number of contour edges. Then we have n c = \Theta( p n ). Pr... |

36 |
Designing the Computational Geometry Algorithms Library CGAL
- Overmars
(Show Context)
Citation Context ...l algorithms [19]. It is intended to provide implementations for the Cgal project, a coordinated effort of seven research groups in Europe for constructing a Computational Geometry Algorithms Library =-=[6, 16, 3]-=-. The approach we are focussing on uses a line sweep algorithm for hidden surface elimination in the projection plane as in the example above for the silhouette computation. A similar line sweep consi... |

36 |
volumes, packing, and protein structure
- Areas
- 1977
(Show Context)
Citation Context ... significantly lower resolution and therefore medium contour edge ratios. mol1 in Figure 12 is a relative crude approximation of the van der Waals surface of a molecule. mol2 is the molecular surface =-=[17]-=-, also known as Conolly surface, of the same molecule. It is obtained from the van der Waals surface by introducing so-called reentrant surface patches. Caveties are removed or flattened as can be see... |

35 |
A survey of object-space hidden surface removal
- Dorward
- 1994
(Show Context)
Citation Context ...sweep, i.e. (O(n + k) log n) where n is the number of vertices and k the number of intersections of edges in the projection. A survey including other hidden surface removal algorithms can be found in =-=[4]-=-. Robustness and exactness are important issues for combinatorial algorithms such as the line sweep. We decided to implement the line sweep with exact arithmetic of bounded but sufficient precision. T... |

19 | Large mesh generation from boundary models with parametric face representation
- Klein, Straber
- 1995
(Show Context)
Citation Context ... 1 to be 4:5, 6:6, and 10:2 for terrain 1, 2, and 3, respectively. The first three examples in Figure 10 were generated by a mesh generation algorithm for objects with parametric face representations =-=[14]-=-. The number in the filename denotes the tolerated error bound in the approximation. The coefficient in c 1 p n is here 21:7 and 30 for cola 0.05 and cola 0.005, respectively. fandisk1 is an optimized... |

17 |
A fast line-sweep algorithm for hidden line elimination
- Nurmi
- 1985
(Show Context)
Citation Context ...r edges is small and the number of intersections of contour edges appears to be even more favorable. The latter is particularly interesting for object space methods based on the line sweep algorithm, =-=[7, 18, 15]-=-. As a specific application we describe the computation of the silhouette of a polyhedral object. This presentation is a preliminary report from a project called CEBaP(Contour Edge Based Polyhedron Vi... |

13 |
Surface reconstruction between simple polygons via angle criteria
- WELZL, WOLFERS
- 1994
(Show Context)
Citation Context ... % nc n int int f int c % intc int kground 10920 240 657 6 7604 2579 369 5 terrain1 800 64 130 16 623 230 49 8 terrain2 3136 128 371 11 3234 1124 201 6 terrain3 12416 256 1139 9 15807 5257 811 5 from =-=[22]-=- have a significantly lower resolution and therefore medium contour edge ratios. mol1 in Figure 12 is a relative crude approximation of the van der Waals surface of a molecule. mol2 is the molecular s... |

12 |
Raster- scan hidden surface algorithm techniques, Sig- graph
- Hnlin, Gear
- 1977
(Show Context)
Citation Context ...r edges is small and the number of intersections of contour edges appears to be even more favorable. The latter is particularly interesting for object space methods based on the line sweep algorithm, =-=[7, 18, 15]-=-. As a specific application we describe the computation of the silhouette of a polyhedral object. This presentation is a preliminary report from a project called CEBaP(Contour Edge Based Polyhedron Vi... |

7 |
Visible feature return at object resolution
- Sequin, Wensley
(Show Context)
Citation Context ...r edges is small and the number of intersections of contour edges appears to be even more favorable. The latter is particularly interesting for object space methods based on the line sweep algorithm, =-=[7, 18, 15]-=-. As a specific application we describe the computation of the silhouette of a polyhedral object. This presentation is a preliminary report from a project called CEBaP(Contour Edge Based Polyhedron Vi... |

3 |
Line-Sweep auf einem Gitter
- Hoffmann
- 1996
(Show Context)
Citation Context ...ecting the sweep line, opens a new region with a changed quantitive invisibility depending on whether the incident facets of the edges already intersect the sweep line or not (for details we refer to =-=[10]-=-). The silhouette is exactly the collection of those parts of the projected contour edges that separate regions of quantitive invisibility zero from those of non zero. They can be reported while the s... |

2 | Gram’s equation – a probabilistic proof
- Welzl
- 1994
(Show Context)
Citation Context ...ar on the boundary of the projection. Since these events are disjoint for all vertices, P 4 i=1 2ff v i is the probability that T projects to a triangle. Using the so-called Gram-Sommerville idendity =-=[9, 21]-=- we can rewrite this sum as 3 2 4 X i=1 ff v i = 2 6 X i=1 ff e i \Gamma 2 : Plugging this into (1), we get Pr[e 1 intersects e 2 ] = 1 2 / 2ff e1 + 2ff e2 \Gamma / 2 6 X i=1 ff e i \Gamma 2 !! : Lemm... |

1 |
A server of public domain 3d objects: http://www.viewpoint.com
- Avalon
(Show Context)
Citation Context ...0 larger than int c . The overall impression is that n seems to dominate all other quantities (apart from int) in most examples. Figure 8. Architecture examples. powerlns and skyscrpr are from Avalon =-=[2]-=-, pagota is from Viewpoint [20], epcot3 is from the Geomview distribution. powerlns skyscrpr pagota epcot3 filename n j@Ej n c % n c n int int f int c % int c int powerlns 9214 0 2500 27 45162 17350 4... |

1 |
Commercial 3d objects and the Avalon server of public domain 3d objects: http://www.viewpoint.com
- DataLabs, UT, et al.
(Show Context)
Citation Context ... of this case to contribute an intersection of front edges is zero. Figure 7. Examples from computer graphics. head is from the Geomview distribution, honda, beethoven, and general are from Viewpoint =-=[20]-=-. head honda beethoven general filename n j@Ej n c % n c n int int f int c % int c int head 3106 58 203 6 2950 456 64 2 honda 13875 574 2998 21 36970 11918 2138 6 beethoven 5461 268 1003 18 9415 2983 ... |

1 |
Equation -- A Probabilistic Proof
- Gram's
- 1994
(Show Context)
Citation Context ...ar on the boundary of the projection. Since these events are disjoint for all vertices, P 4 i=1 2ff v i is the probability that T projects to a triangle. Using the so-called Gram-Sommerville idendity =-=[9, 21]-=- we can rewrite this sum as 3 2 4 X i=1 ff v i = 2 6 X i=1 ff e i \Gamma 2 : Plugging this into (1), we get Pr[e 1 intersects e 2 ] = 1 2 / 2ff e1 + 2ff e2 \Gamma / 2 6 X i=1 ff e i \Gamma 2 !! : Lemm... |