Mathematically consistent alternatives to Euclidean geometry have been developed over the past hundred years. The noneuclidean geometries can be distinguished by the behavior of parallel lines: in Euclidean space there is exactly one line passing through a given point which is parallel to a given line, but in hyperbolic geometry there are many. In this space the area of a circle grows exponentially with respect to its radius, whereas in Euclidean space the area only grows linearly. Thanks to this property of hyperbolic distance we have a convenient way to visualize exponentially growing trees.

The simplest way to draw 3D hyperbolic pictures is in the interior of
a ball. We use Euclidean straight lines, but the way we measure
distance is changed so that the surface of the ball is infinitely far
away from the origin. From the outside, objects very near the center
seem almost Euclidean but seem to grow distorted and smaller as they
are translated towards the surface of the ball. This representation of
hyperbolic space is known as the *projective* or *Klein*
model. Luckily we can draw such pictures using 4 x 4 matrices: since
the standard graphics pipeline uses homogeneous coordinates,
interactive speeds are easily achieved on modern workstations. Figure
3 shows a navigation sequence in the projective model. The
look and feel of the system is difficult to communicate with still
pictures, a video is necessary to do it
justice.

Figure 3: Motion in hyperbolic space (the projective model)

WebOOGL [120 KB] VRML [250 KB] MPEG [900 KB]

The *conformal* or *Poincaré* model is related to the
projective model by a simple transformation. In this model, straight
lines are drawn as arcs and flat faces are drawn as parts of spheres.
The advantage is that angles are always drawn correctly. Unfortunately
we cannot use 4 x 4 matrices to represent motion in the conformal
model, and drawing arcs and curved faces also requires much greater
subdivision than in the projective case. While this model is more
computationally demanding than the projective one, it is within the
reach of most workstations.

If we allow the camera to go outside of the ball we can see the entire space at once. When we constrain the camera itself according to hyperbolic isometries, it can never move outside of the ball and we see what hyperbolic space would look like from the insider's point of view. These three models provide different and useful intuition, so we use whichever of them is appropriate at any given time. Figure 4 shows two levels of the Geometry Center Weblet in the three models. (We use ``Weblet'' to refer to a section of the Web .) Here we have drawn the ``sphere at infinity'' for the two outsider models but in all other pictures the sphere is not shown.

Figure 4: The projective, conformal, and ``insider'' models of hyperbolic space

MPEG projective [1.15MB] MPEG conformal [700KB] MPEG insider [360KB]

There has been considerable work at the Geometry Center on
visualizing hyperbolic geometry. The 1991 mathematical animation *Not Knot* [3] includes a groundbreaking flythrough of hyperbolic
space. A reference on hyperbolic navigation can be found in
[1], which to the best of our knowledge contains the first
recorded suggestion of using hyperbolic space for tree visualization.
See also [2] for more exposition of hyperbolic geometry
aimed at the computer graphics community.

The publicly distributed Geomview 3D viewer [8] includes built-in support for interactive navigation in noneuclidean geometries, including hyperbolic geometry. Details on the implementation of the hyperbolic transformation libraries used in Geomview can be found in [7].

A recent paper from Lamping et al at Xerox PARC [4] has brought hyperbolic visualization to the attention of the computer-human interaction community. They ran user tests of a 2D Poincaré model browser visualizing various kinds of hierarchical information, including organizational charts and a Weblet. Their work on information visualization is confined to the 2D hyperbolic plane, and deals only with acyclic graphs.

Hyperbolic space has a similar visual effect to a fisheye camera lens, used by [10] for information visualization. The fisheye paradigm calls for several ad-hoc decisions, the advantage of hyperbolic geometry is it provides an elegant and powerful mathematical framework for display and navigation.

Tue Nov 21 23:43:05 CST 1995