## Non-Euclidean Spring Embedders (2004)

Citations: | 7 - 2 self |

### BibTeX

@MISC{Kobourov04non-euclideanspring,

author = {Stephen G. Kobourov and et al.},

title = {Non-Euclidean Spring Embedders},

year = {2004}

}

### OpenURL

### Abstract

We present a method by which force-directed algorithms for graph layouts can be generalized to calculate the layout of a graph in an arbitrary Riemannian geometry. The method relies on extending the Euclidean notions of distance, angle, and force-interactions to smooth non-Euclidean geometries via projections to and from appropriately chosen tangent spaces. In particular, we formally describe the calculations needed to extend such algorithms to hyperbolic and spherical geometries.

### Citations

515 | A heuristic for graph drawing
- Eades
- 1984
(Show Context)
Citation Context ...perbolic space and in spherical space, such as those in Fig. 1. 2 Related Work 2.1 Force-Directed Layouts Force-directed algorithms are a well-known and powerful tool for laying out arbitrary graphs [=-=Eades 1984-=-; Fruchterman and Reingold 1991; Kamada and Kawai 1989]. Such methods define an objective function which maps each graph layout into a number in R + representing the energy of the layout. Generally, t... |

457 | An algorithm for drawing general undirected graphs
- Kamada, Kawai
- 1989
(Show Context)
Citation Context ... as those in Fig. 1. 2 Related Work 2.1 Force-Directed Layouts Force-directed algorithms are a well-known and powerful tool for laying out arbitrary graphs [Eades 1984; Fruchterman and Reingold 1991; =-=Kamada and Kawai 1989-=-]. Such methods define an objective function which maps each graph layout into a number in R + representing the energy of the layout. Generally, this energy function is defined in such a way that low ... |

428 | Graph drawing by force-directed placement - Fruchterman, Reingold - 1991 |

336 | A Focus+Context Technique Based on Hyperbolic Geometry for Visualising Large Hierarchies
- Lamping, Rao, et al.
- 1995
(Show Context)
Citation Context ... also been noted that certain nonEuclidean geometries, specifically hyperbolic geometry, have properties which are particularly well suited to the layout and visualization of large classes of graphs [=-=Lamping et al. 1995-=-; Munzner 1997]. We present a method by which a force-directed algorithm can be generalized so that it can compute a graph layout in any of a large class of geometries (known as Riemannian geometries)... |

115 |
H3: Laying out large directed graphs in 3D hyperbolic space
- MUNZNER
- 1997
(Show Context)
Citation Context ... certain nonEuclidean geometries, specifically hyperbolic geometry, have properties which are particularly well suited to the layout and visualization of large classes of graphs [Lamping et al. 1995; =-=Munzner 1997-=-]. We present a method by which a force-directed algorithm can be generalized so that it can compute a graph layout in any of a large class of geometries (known as Riemannian geometries), so long as t... |

104 | Fundamentals of spherical parameterization for 3d meshes - Gotsman, Gu, et al. - 2003 |

79 | A fast multi-scale method for drawing large graphs
- Harel, Koren
- 2000
(Show Context)
Citation Context ...force-directed algorithms work well for small graphs, recently such algorithms have been extended to deal with graphs with hundreds of thousands of vertices using multi-scale and spectral techniques [=-=Harel and Koren 2002-=-; Koren et al. 2002]. Spring embedders thus far have been restricted to ndimensional Euclidean space. This restriction is due in part to the simplicity of the algorithms when formulated in Euclidean s... |

72 | Visualizing the structure of the World Wide Web in 3D hyperbolic space
- MUNZNER, BURCHARD
- 1995
(Show Context)
Citation Context ...nd 140 edges. method is inapplicable to certain geometries (e.g., hyperbolic geometry). 2.2 Hyperbolic Graph Drawing Much of the work on non-Euclidean graph drawing has been done in hyperbolic space [=-=Munzner and Burchard 1996-=-; Munzner 1997] which offers certain advantages over Euclidean space. For example, in hyperbolic space it is possible to compute a layout for a complete tree with both uniform edge lengths and uniform... |

62 | Ace: A fast multiscale eigenvectors computation for drawing huge graphs - Koren, Harel - 2002 |

43 | Drawing large graphs with H3Viewer and site manager (system demonstration
- Munzner
- 1998
(Show Context)
Citation Context ... applications calculate the layout of a general graph using this method, the layout is calculated using a spanning tree of the graph and the extra edges are then added in without altering the layout [=-=Munzner 1998-=-]. This method works well for tree-like and quasi-hierarchical graphs, or for graphs where domain-specific knowledges provides a way to create a meaningful spanning tree. However, for general graphs (... |

42 | Graphael: Graph animations with evolving layouts
- Erten, Harding, et al.
- 2004
(Show Context)
Citation Context ...tangent space. x y Figure 8: Mapping from the tangent space via a rotation. 6 Example Layouts in H 2 and S 2 In Figures 9-11 we consider a title-word graph obtained from the graph drawing literature [=-=Erten et al. 2003-=-]. This graph has 27 nodes and 50 edges and the nodes correspond to titlewords from papers in the 1999 Graph Drawing conference. The size of a node is determined by its frequency and edges are placed ... |

36 | A multi-dimensional approach to force-directed layouts of large graphs - Gajer, Goodrich, et al. - 2000 |

32 | Self-Organizing Maps on non-euclidean Spaces - Ritter - 1999 |

11 | Some three-dimensional graph drawing algorithms
- Ostry
- 1996
(Show Context)
Citation Context ...tructure of Euclidean space with well-defined notions of distances and angles. Some work, however, has been done on constraining force-directed algorithms to the surface of three-dimensional objects [=-=Ostry 1996-=-]. This work is based on a differential equation formulation of the motion of the nodes in the graph, and is flexible in that it allows a layout on almost any object, even multiple objects. Since the ... |

5 | Hyperbolic self-organizing maps for semantic navigation - Ontrup, Ritter - 2001 |

5 | Hyperbolic geometry, MÃ¶bius transformations, and geometric optimization, 2003 - Eppstein |