At 12:21 PM 9/13/01 -0400, Spencer Simpson wrote:
>Well, yes and no. As the network space is a series of lines,
>a "network circle" is a collection of isolated points.
>
>Mr. Brooks may be interested in a figure arrived at when the network
>is embedded in a two-dimensional Euclidean space (i. e. it is displayed
>on a map) and the network "circle" points are connected to each other
>with lines across the new space. The lines do not serve any analytical
>purpose; they exist for cosmetic purposes only.
I agree with this comment, with some modification, because those cosmetic
lines can be misleading in maps of sparse networks.
First, nobody claimed a network circle was connected. (Indeed, an ordinary
circle in one-dimensional Euclidean space always consists of two
disconnected points.) That is not a requirement.
Second, and more to the point, it is indeed fruitful to contemplate the
network within its two-dimensional Euclidean (or spherical) space. It is
from that space the network inherits its metric, after all. But one can go
further toward a realistic model by conceptually extending the network
metric to an open set of points surrounding it.
The "distance" from point p to point q, for any points p and q in a
neighborhood of the network (but not necessarily on it) might be defined as
a sum of three terms: the shortest distance from p to the network, the
shortest distance from q to the network, and the network distance from a
point nearest p to a point nearest q. (This can get you into trouble when
you think about how to find these nearest points, because in many cases
they are not unique.)
Even better--this works well in raster-based GIS's like Idrisi, GRASS, or
Spatial Analyst--you define the length of any path from p to q as the cost
of traveling from p to q along any route (on the network or not). The cost
equals the network distance where the travel occurs along the network and
otherwise is some large multiple (greater than 1.0) of the distance. If
the network is not too convoluted and the multiple is large enough, then
this cost effectively extends the definition of network distance to all
points within a neighborhood of the network. The "circles" in this metric
look a lot like buffers around the network segments, which can lead to an
interesting and informative map display.
Another approach is the cartogram: distort the network on the map so that
all points on a network circle lie on a common Euclidean circle. This
could be an effective way of drawing, say, a subway map: the station would
be at the center and the radial map distance to any other station could
indicate its actual distance along the subway lines.
I think there's plenty of room here for cartographic creativity and
imagination...
--Bill Huber
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