## A Framework for Drawing Planar Graphs with Curves and Polylines (1998)

Venue: | J. Algorithms |

Citations: | 14 - 3 self |

### BibTeX

@INPROCEEDINGS{Goodrich98aframework,

author = {Michael T. Goodrich and Christopher G. Wagner},

title = {A Framework for Drawing Planar Graphs with Curves and Polylines},

booktitle = {J. Algorithms},

year = {1998},

pages = {153--166},

publisher = {Springer}

}

### Years of Citing Articles

### OpenURL

### Abstract

We describe a unified framework of aesthetic criteria and complexity measures for drawing planar graphs with polylines and curves. This framework includes several visual properties of such drawings, including aspect ratio, vertex resolution, edge length, edge separation, and edge curvature, as well as complexity measures such as vertex and edge representational complexity and the area of the drawing. In addition to this general framework, we present algorithms that operate within this framework. Specifically, we describe an algorithm for drawing any n-vertex planar graph in an O(n) O(n) grid using polylines that have at most two bends per edge and asymptotically-optimal worstcase angular resolution. More significantly, we show how to adapt this algorithm to draw any n-vertex planar graph using cubic Bézier curves, with all vertices and control points placed within an O(n) O(n) integer grid so that the curved edges achieve a curvilinear analogue of good angular resolution. Al...

### Citations

244 |
Algorithms for drawing graphs: An annotated bibliography
- Eades, Tamassia
- 1989
(Show Context)
Citation Context ... a drawing is dened as the area of the smallest rectangle that encloses the drawing. We typically desire the area of a drawing to be small while not violating our aesthetic goals. Di Battista et al. [4] and Tamassia [14] summarize several known bounds on area-tradeos for various aesthetic properties for dierent types of drawings. For example, (n 2 ) worst-case area is required for planar polyline... |

238 | Efficient planarity testing - Hopcroft, Tarjan - 1974 |

233 | A technique for drawing directed graphs
- Gansner, Koutsofios, et al.
- 1993
(Show Context)
Citation Context ...hat allow for curves and polyline smoothing, but we have not been able tosnd much written on formal studies of curve drawings. A notable exception to this is work of Gansner, Koutsoos, North, and Vo [=-=6]-=- that addresses the issue of curve drawings for general graphs, considering aesthetic criteria such as the hierarchical structure of the underlying graph, the edge length, symmetry, and the number of ... |

199 |
Embedding planar graphs on the grid
- Schnyder
- 1990
(Show Context)
Citation Context ...4] summarize several known bounds on area-tradeoffs for various aesthetic properties for different types of drawings. For example, Θ(n2 ) worst-case area is required for planar polyline grid drawings =-=[2, 3, 10, 12]-=-. Another interesting example of a tradeoff between the aesthetic criteria and the complexity measures is the angular resolution of the drawing. Clearly, the angular resolution of any graph drawing is... |

156 |
How to draw a planar graph on a grid
- Fraysseix, Pach, et al.
- 1990
(Show Context)
Citation Context ...e realistically achieved. Examples of such studies are too numerous to enumerate, but some well-known examples include the importance of the drawing area identified by de Fraysseix, Pach, and Pollack =-=[3]-=-, the illumination of the value of bend minimization in polyline drawings by Tamassia [13], the discussion of aspect ratio by Chan et al. [1], and the identification of angular resolution as a signfic... |

137 |
On embedding a graph in the grid with the minimum number of bends
- Tamassia
- 1987
(Show Context)
Citation Context ...e well-known examples include the importance of the drawing area identified by de Fraysseix, Pach, and Pollack [3], the illumination of the value of bend minimization in polyline drawings by Tamassia =-=[13]-=-, the discussion of aspect ratio by Chan et al. [1], and the identification of angular resolution as a signficant aesthetic criterion by Formann et al. [5], and its further study by Kant [10], Garg an... |

68 | Drawing planar graphs using the canonical ordering
- Kant
- 1996
(Show Context)
Citation Context ...Tamassia [13], the discussion of aspect ratio by Chan et al. [1], and the identification of angular resolution as a signficant aesthetic criterion by Formann et al. [5], and its further study by Kant =-=[10]-=-, Garg and Tamassia [7], and Malitz and Papakostas [11]. ⋆ Work by this author is supported in part by the U.S. Army Research Office under Grant DAAH04-96-1-0013 and by NSF under Grant CCR-96-25289. ⋆... |

64 | How to draw a planar graph on a grid, Combinatorica 10 - Fraysseix, Pach, et al. - 1990 |

37 | A linear-time algorithm for drawing planar graphs
- Chrobak, Payne
- 1995
(Show Context)
Citation Context ...Pollack [3] for drawing an n-vertex planar graph in an O(n) O(n) integer grid using straight line segments to represent edges. Originally presented as an O(n log n) time algorithm, Chrobak and Payne [=-=2]-=- show how to cleverly use a tree data structure to reduce the running time of this algorithm to O(n). This algorithm does not achieve good angular resolution, however, because edges incident on the sa... |

37 | Algorithms for Drawing Planar Graphs - Kant - 1993 |

35 | On the angular resolution of planar graphs - Malitz, Papakostas - 1994 |

30 |
Drawing graphs in the plane with high resolution
- Formann, Hagerup, et al.
- 1993
(Show Context)
Citation Context ...minimization in polyline drawings by Tamassia [13], the discussion of aspect ratio by Chan et al. [1], and the identication of angular resolution as a signcant aesthetic criterion by Formann et al. [5=-=-=-], and its further study by Kant [10], Garg and Tamassia [7], and Malitz and Papakostas [11]. This paper is directed at continuing this tradition by articulating a unied framework for describing aesth... |

25 | Planar drawings and angular resolution: Algorithms and bounds
- Garg, Tamassia
- 1994
(Show Context)
Citation Context ...cussion of aspect ratio by Chan et al. [1], and the identication of angular resolution as a signcant aesthetic criterion by Formann et al. [5], and its further study by Kant [10], Garg and Tamassia [7=-=-=-], and Malitz and Papakostas [11]. This paper is directed at continuing this tradition by articulating a unied framework for describing aesthetic criteria and complexity measures for drawings of graph... |

18 | Optimizing area and aspect ratio in straight-line orthogonal tree drawings
- Chan, Goodrich, et al.
- 1997
(Show Context)
Citation Context ... drawing area identied by de Fraysseix, Pach, and Pollack [3], the illumination of the value of bend minimization in polyline drawings by Tamassia [13], the discussion of aspect ratio by Chan et al. [=-=-=-1], and the identication of angular resolution as a signcant aesthetic criterion by Formann et al. [5], and its further study by Kant [10], Garg and Tamassia [7], and Malitz and Papakostas [11]. This ... |

14 | Graph Drawing
- Tamassia
- 1997
(Show Context)
Citation Context ...ned as the area of the smallest rectangle that encloses the drawing. We typically desire the area of a drawing to be small while not violating our aesthetic goals. Di Battista et al. [4] and Tamassia =-=[14]-=- summarize several known bounds on area-tradeoffs for various aesthetic properties for different types of drawings. For example, Θ(n2 ) worst-case area is required for planar polyline grid drawings [2... |