## Finding the Best Viewpoints for Three-Dimensional Graph Drawings (1997)

Venue: | Proc. 5th International Symp. on Graph Drawing (GD ’97 |

Citations: | 18 - 0 self |

### BibTeX

@INPROCEEDINGS{Eades97findingthe,

author = {Peter Eades and Michael E. Houle and Richard Webber},

title = {Finding the Best Viewpoints for Three-Dimensional Graph Drawings},

booktitle = {Proc. 5th International Symp. on Graph Drawing (GD ’97},

year = {1997},

pages = {87--98},

publisher = {Springer}

}

### OpenURL

### Abstract

In this paper we address the problem of finding the best viewpoints for three-dimensional straight-line graph drawings. We define goodness in terms of preserving the relational structure of the graph, and develop two continuous measures of goodness under orthographic parallel projection. We develop Voronoi variants to find the best viewpoints under these measures, and present results on the complexity of these diagrams.

### Citations

1760 |
Computational Geometry: An Introduction
- Preparata
- 1985
(Show Context)
Citation Context ...ers the significant bad viewpoints of Kamada and Kawai, and leads to faster algorithms than those in [4]. Bose et al. suggest an algorithm to describe all bad viewpoints in by building an arrangement =-=[5,19]-=- of curves consisting of bad viewpoints. Under their model this requires O(jEj 3 log jEj + k 0 ) time, where the parameter k 0 is O(jEj 6 ) in the worst case. In Sec. 4 we employ the same technique wi... |

565 | Voronoi diagrams – a survey of a fundamental geometric data structure
- Aurenhammer
(Show Context)
Citation Context ...o points each. tional separation, and in Sec. 5.1 we show how to find all best viewpoints under rotational separation using the rotational separation diagram. This is a variant of the Voronoi diagram =-=[2]-=- which can be built in O((j\Psi j +k) log(j\Psi j +k)) time, with j\Psi j and k as above. In Sec. 5.2 we define goodness to be the observed separation after projection and describe the observed separa... |

333 |
A sweepline algorithm for Voronoi diagrams
- Fortune
- 1986
(Show Context)
Citation Context ...d the rotational separation diagram by modifying a planar algorithm for the Voronoi diagram of line-segments to work on the sphere S 2 . The similar problem of modifying Fortune's sweepline algorithm =-=[13]-=- for a cone has been described by Dehne and Klein [7]. The rotational separation diagram has O(jSj) size and requires O(jSj log jSj) time to build, where jSj is the size of the bad viewpoint arrangeme... |

255 |
Optimal Search in Planar Subdivision
- Kirkpatrick
- 1983
(Show Context)
Citation Context ...ion diagram, this time using the Voronoi vertices of the first diagram as the sites. This requires the same preprocessing time and space as before, after which we can apply a point-location algorithm =-=[15,19]-=-, again with the same preprocessing time and space requirements. The resulting diagram can be used to find the nearest site (the locally best viewpoint) in O(log jSj) time. Result 2. The rotational se... |

124 | Evaluating stereo and motion cues for visualizing information nets in three dimensions
- Ware, Franck
- 1996
(Show Context)
Citation Context ...o-dimensional image: finding a good viewpoint. There is some experimental evidence that three-dimensional graph drawings have advantages over their two-dimensional counterparts. One claimed advantage =-=[23]-=- is that three dimensions allow users to work with larger graphs -- the natural three-dimensional actions of rotation and translation allow users to resolve ambiguities in large graphs while maintaini... |

72 | Visualizing the structure of the World Wide Web in 3D hyperbolic space
- MUNZNER, BURCHARD
- 1995
(Show Context)
Citation Context ...many areas of computing. For example: to visualise the structures of object-oriented [16,24] and parallel [22] software; for interactive information retrieval [11]; and to navigate the World-Wide Web =-=[18,20]-=-. There has been much work on creating graph drawings that are both easy to remember and easy to understand [8]. Most work has been to produce twodimensional drawings, but recent work addresses three-... |

60 |
Preserving the mental map of a diagram
- Eades, Lai, et al.
- 1991
(Show Context)
Citation Context ...low users to work with larger graphs -- the natural three-dimensional actions of rotation and translation allow users to resolve ambiguities in large graphs while maintaining their overall mental map =-=[9]-=-. Ware et al. [23] consider the problem of finding connections in a threedimensional graph drawing, with a variety of display and navigation combinations. Using a sequence of human experiments, they s... |

46 |
A simple method for computing general position in displaying three-dimensional objects
- Kamada, Kawai
- 1988
(Show Context)
Citation Context ...then it moves the viewpoint to be perpendicular to the surface. In Sec. 2 we briefly present some background material, and outline previous work specifically on the problem of finding good viewpoints =-=[4,14]-=-. In Sec. 3 we model good viewpoints in terms of the information conveyed by a drawing, then in Sec. 4 we use the techniques described by Bose, Gomez, Ramos and Toussaint [4] to find all bad viewpoint... |

34 |
Intersection and decomposition algorithms for planar arrangements
- Agarwal
- 1991
(Show Context)
Citation Context ...ase size complexity of the restricted observed separation diagram ( E = ;) on jSj point sites lies within the bounds\Omega (jSj 2 ) and O(jSj 2 2 ff(jSj) ), where ff is the inverse Ackermann function =-=[1]-=-. Proof. An example of a diagram with\Omega (jSj 2 ) size is given in Fig. 7. It is a simple adaption of Aurenhammer and Edelsbrunner's [3] worst case example for the multiplicatively-weighted Voronoi... |

29 |
Knot theory
- Livingston
- 1996
(Show Context)
Citation Context ...drawing (V 0 = ;), excluding viewpoints for which edges appear collinear. Bose et al. [4] preserve the depth-order of a wire-frame drawing, permitting only viewpoints that yield regular projections 1 =-=[17]-=-. In Sec. 3, we define good viewpoints to be those that preserve the abstract graph of a drawing, where a drawing may include isolated vertices. Our definition covers the significant bad viewpoints of... |

19 |
An application of three-dimensional visualization to objectoriented programming
- Koike
- 1992
(Show Context)
Citation Context ...results on the complexity of these diagrams. 1 Introduction Three-dimensional drawings of graphs are being used in many areas of computing. For example: to visualise the structures of object-oriented =-=[16,24]-=- and parallel [22] software; for interactive information retrieval [11]; and to navigate the World-Wide Web [18,20]. There has been much work on creating graph drawings that are both easy to remember ... |

14 |
Tamassia and I. Tollis, "Algorithms for Drawing Graphs: An Annotated Bibliography
- Battista, Eades, et al.
- 1994
(Show Context)
Citation Context ...ware; for interactive information retrieval [11]; and to navigate the World-Wide Web [18,20]. There has been much work on creating graph drawings that are both easy to remember and easy to understand =-=[8]-=-. Most work has been to produce twodimensional drawings, but recent work addresses three-dimensions [6]. Much of this work has concentrated on the problems of finding a good three-dimensional drawing ... |

7 | Graph-drawing contest report
- Eades, Marks, et al.
- 1997
(Show Context)
Citation Context ...ree-dimensional graph drawing must be rendered as a static two-dimensional image. Using a good viewpoint under these circumstances can be critical, as was evident in last year's graph drawing contest =-=[10]-=-. Some existing systems already support simple versions of logical selection. The application ivview [21] navigates around three-dimensional scenes. It has a logical selection function that allows use... |

3 |
3D Interactive Animation for Visualization of Parallel Design
- Trichina, Thomas
- 1995
(Show Context)
Citation Context ...xity of these diagrams. 1 Introduction Three-dimensional drawings of graphs are being used in many areas of computing. For example: to visualise the structures of object-oriented [16,24] and parallel =-=[22]-=- software; for interactive information retrieval [11]; and to navigate the World-Wide Web [18,20]. There has been much work on creating graph drawings that are both easy to remember and easy to unders... |

2 |
Edelsbrunner: "An Optimal Algorithm for Constructing the Weighted Voronoi Diagram
- Aurenhammer, H
- 1984
(Show Context)
Citation Context ...rangement; however its bisectors are more complex, being defined by the equation \Delta(sd; /(a; b)) = \Delta(sd; /(a 0 ; b 0 )). This form is similar to the multiplicatively-weighted Voronoi variant =-=[2,3]-=-, but with an extra sine factor. Consider the restricted case where E = ;. Each vertex-vertex occlusion /(v i ; v j ) generates a point Voronoi site s a =sd ij with a corresponding weight w a = ae(v i... |

2 |
Toussaint: "Drawing Nice Projections of Objects in Space
- Bose, Gomez, et al.
- 1995
(Show Context)
Citation Context ...then it moves the viewpoint to be perpendicular to the surface. In Sec. 2 we briefly present some background material, and outline previous work specifically on the problem of finding good viewpoints =-=[4,14]-=-. In Sec. 3 we model good viewpoints in terms of the information conveyed by a drawing, then in Sec. 4 we use the techniques described by Bose, Gomez, Ramos and Toussaint [4] to find all bad viewpoint... |

2 |
Ruskey: "Three-Dimensional Graph Drawing
- Cohen, Eades, et al.
- 1996
(Show Context)
Citation Context ... been much work on creating graph drawings that are both easy to remember and easy to understand [8]. Most work has been to produce twodimensional drawings, but recent work addresses three-dimensions =-=[6]-=-. Much of this work has concentrated on the problems of finding a good three-dimensional drawing for a graph. In this paper, we consider a problem in the process of rendering a three-dimensional graph... |

2 |
The Voronoi Diagram of Points on a Cone
- Dehne, Klein
(Show Context)
Citation Context ...nar algorithm for the Voronoi diagram of line-segments to work on the sphere S 2 . The similar problem of modifying Fortune's sweepline algorithm [13] for a cone has been described by Dehne and Klein =-=[7]-=-. The rotational separation diagram has O(jSj) size and requires O(jSj log jSj) time to build, where jSj is the size of the bad viewpoint arrangement that defines its Voronoi sites. The size jSj is O(... |

2 |
VIRI: "Actual Listing of Information Visualization Systems
- FADIVA
- 1995
(Show Context)
Citation Context ...nal drawings of graphs are being used in many areas of computing. For example: to visualise the structures of object-oriented [16,24] and parallel [22] software; for interactive information retrieval =-=[11]-=-; and to navigate the World-Wide Web [18,20]. There has been much work on creating graph drawings that are both easy to remember and easy to understand [8]. Most work has been to produce twodimensiona... |

2 |
Information Diagrams for the DOOMed Generation
- Regan
- 1996
(Show Context)
Citation Context ...many areas of computing. For example: to visualise the structures of object-oriented [16,24] and parallel [22] software; for interactive information retrieval [11]; and to navigate the World-Wide Web =-=[18,20]-=-. There has been much work on creating graph drawings that are both easy to remember and easy to understand [8]. Most work has been to produce twodimensional drawings, but recent work addresses three-... |

2 |
Franck: "Visualizing Object Oriented
- Ware, Hui, et al.
- 1993
(Show Context)
Citation Context ...results on the complexity of these diagrams. 1 Introduction Three-dimensional drawings of graphs are being used in many areas of computing. For example: to visualise the structures of object-oriented =-=[16,24]-=- and parallel [22] software; for interactive information retrieval [11]; and to navigate the World-Wide Web [18,20]. There has been much work on creating graph drawings that are both easy to remember ... |