Intersection is probably not the ideal operation, since it could exclude a page very close to one focus but a little too far from the other, while including a page the maximum distance from each focus. Instead, we probably want to take the set of points where the sum of their distances from the foci is less than a constant; in other words, an ellipse.* We used our system to find empirically that paths from Yahoo's CS and public-interest indices met, appropriately, at Electronic Privacy Information Center. While we have defined the distance between two pages as the shortest path between them, other possible definitions would take into account the number of paths between two pages.
Crawling on demand also allows finding information that has not yet been indexed by other crawlers. For example, when searching for information on Cecilia Rodriguez, we were given a URI for a page that no longer existed. We removed text from the right end of the URI until we found a page that still existed. By expanding links from that page, we were able to find not only the moved page but also information on Cecilia Rodriguez that had not yet been indexed. Such searches can be facilitated by preferentially expanding pages or links containing relevant text.
Another application of link geometry is discovering the class of a page from its link structure. For example, a homepage would tend to contain links downward in the file hierarchy with perhaps a few upwards and sideways, while an index would include a large number of links to many different sites. A clue to the category of a page can be found by walking backwards up links that point to it until the first Yahoo page, for example, is reached and looking at the header. Another application of back links is automatically finding pages (indices) that point to pages you like; expanding from the index may help you discover more such pages.