At 11:07 AM 9/13/01 +0100, Brooks, Stephen wrote:
>I
>would like to know the name of a line that links all points of equal
>distance from a specific point. This is not necessarily just a circle
>around the point if you take into consideration permeability of route
>networks.
"Isometric curve" would be correct but I have never seen it used.
The proper mathematical term really is "circle." It applies in any setting
with a few simple features:
1. You have any set S. Call its elements "points".
2. There is a real-valued function d(p,q) defined for every pair of
points p and q, such that
2a. d(p, q) >= 0 and d(p, q) = 0 if and only if p =
q. ("Positive definiteness")
2b. d(p, q) = d(q, p) ("Symmetry")
2c. For all points p, q, r, d(p,q) + d(q,r) >= d(p,r)
("Triangle inequality")
When properties 2a-c hold, you are allowed to call d a "distance" and the
pair (S, d) is called a "metric space." The definition of metric space
captures the essential properties of all distances and, despite its
simplicity, leads to a rich and deep theory.
In any metric space, the circle of radius a about a point p is defined to
be the set of all points q such that d(p,q) = a.
The distance defined by shortest routes on a *symmetric* network makes all
points accessible to that network into a metric space. That is easy to
see, because 2a is inherited from the usual distance function, 2b is what I
mean by "symmetric" (all networks without one-way streets are symmetric),
and 2c follows from the shortest-route idea ("shortest" implies d(p,r) can
be no greater than the distance traveled from p to r when visiting any
intermediate point q; writing that down gives 2c).
The terminological trick, then, is to specify what metric you mean:
Euclidean, spherical, or network. The phrase, "circle in the network
metric," or "network circle" for short, would be appropriate for the
latter. This gives us three kinds of circles, respectively: (ordinary)
circles, geodesic circles, and network circles.
--Bill Huber
_______________________________________________________________________
List hosting provided by Directions Magazine | www.directionsmag.com |
To unsubscribe, send e-mail to [EMAIL PROTECTED] and
put "unsubscribe MapInfo-L" in the message body.
