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The graph can be easily laid out in the hyperbolic plane using uniform edge lengths. Indeed, this property of hyperbolic geometry is one of the motivations behind its use. However, we would like to optimize our use of space so that as many generations as possible are visible. (This optimization also forestalls a bit longer the inevitable cumulative floating-point error). So we would like edges that connect nodes to be as short as possible, yet we must also avoid overlaps between the cone trees of all future generations. In order to most economically meet these constraints, the hyperbolic length of the edge connecting nodes and should be

where denotes the smallest angle between edges incident on node (see Figure 5).

When the cone tree angle (a user-specifiable parameter) is 90 degrees, cones become flat disks. While this would negate most of the advantages of cone trees in Euclidean space, in hyperbolic space disks are very convenient. The hierarchy of disks has no directional bias: locally, child nodes are indistinguishable from parent nodes. Whichever node is closest to the origin of the sphere ``feels'' like the root node.


Figure 5: Angle between edges incident on node i.

Tamara Munzner
Tue Nov 21 23:43:05 CST 1995