A Hyperbolic Image Viewer

Tom White, April 1999.

Abstract

This page is devoted to a discussion of the benefits hyperbolic displays can bring to visualisation of large datasets. I have provided some Java Applets to demonstrate some of the ideas.

Java Applets

Escher (applet), (source image), (hyperbolic image)
Stoke Newington (applet), (source image), (hyperbolic image)
Windows Desktop (applet), (source image), (hyperbolic image)

2019 Update: Applets are no longer supported in browsers, so to run, install a JDK and type:
appletviewer escher.html, or
appletviewer stokie.html, or
appletviewer desktop.html.

Hyperbolic Geometry in Computer Displays

Although steadily becoming cheaper screen 'real estate' is still at a premium. Computer applications developers strain to squeeze as much information as possible from each patch of pixels while monitor manufacturers grow their models inexorably - but slowly. Until we have wall-sized display units that allow us to shuffle views as flexibly as we hang pictures now we shall continue to use nifty space-saving measures such as scrolling panels and switchable windows.

It may then come as a surprise that - in the world of pure mathematics at least - an infinite amount of space can be contained in a finite area. The trick of course is that the visible size of objects in such a space, called a hyperbolic geometry, decreases to nothing as you get closer to the 'edge'. To illustrate this we can use the Poincare/ Model which is one such hyperbolic geometry. This model represents a point in hyperbolic space by a point on the unit disk - that is, the numbers in the complex plane with magnitude less than 1. Distances in this disk are not measured using the familiar Euclidean metric, but rather one which attributes the distance to the edge of the disk as infinity. This means that the whole Euclidean plane is 'contained' in the unit disk. The only problem is the edge is a little squashed.

Nice things about this new view:

• There is no artificial cut-off. In the scrollbar model there is a fixed rectangular area which acts as an arbitrarily sized clipping area. And the size of this area is often dictated by external concerns, such as the left over space in a given window. The scrollbars are imposed for practical reasons - not because they are an integral part of the view. The hyperbolic view has no such unnatural constructs.
• Cut-off is not sharp. You do not know what is just beyond the scrollbar. In a hyperbolic view, you have at least some visual indication - a gestalt - of what is coming up. This hyperbolic method thus lends itself well to applications where there is some form of connectivity between regions of space, such as maps. Applications with unordered, linear, or text only elements don't benefit as well from a hyperbolic projection; list boxes for example.
• The view can fit in any fixed display space. Infinite Euclidean space - the usual (unstated) display model - can be wrapped in a disk of arbitrary size. Although only the central portion, roughly three-quarters of the display area of the disk, is clearly discernible - in a sense the view takes in the whole object. The display can mask the size of the object (although this can be a disadvantage) which brings a certain elegant simplicity.
• Natural rotation. In scroll panel implementations there is no obvious mechanism for rotation, and if there is then the height and width of the display object changes which can be disconcerting or annoying. Since the hyperbolic view disguises absolute size and since the display is a circular presentation, rotation is an entirely natural motion.
• Controls neatly occupy the space around the disk.

But:

• There is no indication of how big the display object is. There is no 'width of scrollbar thumb' equivalent measure.

Promising applications include:

• Maps, both geographic and topological. With clear directional markings navigation - by dragging and rotation, or programmatically - is very intuitive. Indeed, 'fly by wire' views appear very promising and could find interesting application in car navigation units (where the driver needs a large magnified view of his or her current location and but only an indication of the geography much further ahead).
• Hierarchies such as trees and graphs. See paper on displaying the WWW in hyperbolic space.
• Tiling of the plane. Since you can 'scroll forever' a nice application would visualising an infinite tiling such as wallpaper (periodic) or a Penrose tiling (aperiodic). See the alhambra project for more on aperiodic tilings.

References

A Focus+Context Technique Based on Hyperbolic Geometry for Visualizing Large Hierarchies by John Lamping, Ramana Rao, and Peter Pirolli.
Visualizing the Structure of the World Wide Web in 3D Hyperbolic Space by Tamara Munzner and Paul Burchard.