11.3 Graph-based Fish Eye Views

Sarkar and Brown [Sar92] extend the fish eye view concept to the graphical display of information in the form of graphs. In Furnas' original scheme one node in a graph would be either shown or not shown at its original location. In Sarkar and Brown's approach nodes can be differently sized and edges stretched depending on the importance of the node in the whole graph. They generalize Furnas' DOI-function for a node by introducing a Visual Worth VW, for a node x. Each node in the graph can be shifted from its original position and changed in size to magnify areas close to the focus and demagnify areas farther away. To define the position of each node in the fish eye view, the following formalism is introduced:

  1. The position Pfisheye of a node in the fish eye view can be expressed as a function of the normal (non-fish eye) position of the node Pnorm and the normal position of the focus Pnorm focus:
  2. Pfisheye= F1 (Pnorm, Pnorm focus)
  3. The size of a node in the fish eye view Sfisheye is a function of its normal size Snorm and normal position Pnorm, its a priori importance API, and the normal position of the focus Pnorm focus:
  4. Sfisheye = F2 (Snorm , Pnorm, Pnorm focus, API )
  5. The amount of detail DTLfisheye that can be displayed for a node depends on the size of the node in the fish eye view Sfisheye and the maximum detail that can be shown DTLmaximum:
  6. DTLfisheye = F3 (Sfisheye , DTLmaximum )
  7. The visual worth VW of a node, which decides whether a node will be shown or not, depends on the distance between the node and the focus in normal coordinates Dnorm and on the a priori importance API of the node.
  8. VW = F4 (Dnorm , API )
To generate useful views, the functions F1, F2, F3, and F4 have to be chosen appropriately. Sarkar and Brown present a prototype implementation where they use the following function F1 for the mapping of the Cartesian coordinates of a point P(x,y):

The x and y coordinates are treated completely independently in the above mapping. Dmax is the distance of the boundary of the window from the focus. The constant d is called the distortion factor. If d=0, then the normal and the fish eye coordinates are the same. The larger the distortion factor d is, the more the nodes are shifted away from the focus and compressed near the boundaries of the graph.


Figure I.56 An undistorted graph (ExFig. 3 from [Sar92])

Figure I.56 shows an undistorted symmetric graph. The graph at the top of figure I.57 displays a fish eye view of the same graph using the Cartesian transformation function F1 described above. In addition, a size transformation function F2 has also been applied.

Figure I.57 A graphical fish eye view using cartesian transformation (top) and using polar transformation (bottom) (d=4) (ExFig. 4 and 5 from [Sar92])

Because early users of the system complained about unnatural proportions of the transformed graph for familiar objects such as geographical maps, Sarkar and Brown implemented a second transformation based on the polar coordinate system. The graph at the bottom of figure I.57 displays the polar transformation of the symmetric graph of figure I.56.

Figure I.58 shows an undistorted graph containing some major cities of the United States at their geographical locations. Figure I.59 demonstrates the same graph in a fish eye view with St. Louis as its focus.


Figure I.58 The initial layout of a graph showing some US cities (ExFig. 1 from [Sar92])

Figure I.59 A fish eye view of the above graph (ExFig. 2 from [Sar92])

While Furnas can be credited with introducing the general idea of fish eye views, the main contribution of Sarkar and Brown is the introduction of a mathematically sound formalism that can be used to generate arbitrary graph-based fish eye views.

Obviously fish eye views offer an excellent means for quickly getting a personalized overview of large information spaces. In that respect, they refine the concept of the overview map. They are somewhat limited in that they need a focus of interest, i.e., a point in the view to whom all other points are put in relation. But on the other hand, this limitation is also their strength because users are very often interested in focused information. Fish eye views are a promising navigation tool offering a mechanism that can be applied to many navigation and information exploration problems in various fields.