🚀 go-pugleaf

In article <19692.2b617217@ecs.umass.edu> padmanab@ecs.umass.edu writes:
>Suppose you have two maps of USA of different 
>scales. Now you place the smaller map inside 
>the bigger map such that it falls completely 
>within it. 

This is possible only if you don't include Alaska and Hawaii, and
consider only the continental landmass that lies between Canada and
Mexico.

>
>Condider each point on the map to be a distinct 
>city!! 
>
>QUESTION: Are there any city/cities on the 
>smaller map which will coincide with the 
>city/cities  on the bigger map? If so why? 
>

Yes, if the larger map can be regarded as a projection of the smaller one
-- this causes the point of projection to represent the same city in both
maps. (Assumption : Alaska and Hawaii have not been considered.)

>-Prasad Padmanabhan
>Dept. of ECE
>University of Massachusetts
>Amherst, MA-01003

Kanad Chakraborty

In article <1993Jan24.154526.24491@zip.eecs.umich.edu> kanad@quip.eecs.umich.edu (Kanad Chakraborty) writes:
>In article <19692.2b617217@ecs.umass.edu> padmanab@ecs.umass.edu writes:
>>Suppose you have two maps of USA of different 
>>scales. Now you place the smaller map inside 
>>the bigger map such that it falls completely 
>>within it. 
>
>This is possible only if you don't include Alaska and Hawaii, and
>consider only the continental landmass that lies between Canada and
>Mexico.

Why? Let one map be, say, a mile across, and the other about a foot across. The
second map will probably fit entirely within, say, Nebraska on the large map.

>>Condider each point on the map to be a distinct 
>>city!! 
>>
>>QUESTION: Are there any city/cities on the 
>>smaller map which will coincide with the 
>>city/cities  on the bigger map? If so why? 
>
>Yes, if the larger map can be regarded as a projection of the smaller one
>-- this causes the point of projection to represent the same city in both
>maps. (Assumption : Alaska and Hawaii have not been considered.)

No special assumptions are needed. It does no harm if you decide to place the
small map within Alaska or Hawaii on the large map.

Pick a city a1 on the large map, and find the corresponding city b1 on the
small map. Since the small map lies within the large map, b1 overlies a point
(city) a2 on the large map, which corresponds to a point b2 on the small map.
Continuing the process, we arrive at a sequence of points a1, a2, a3, ... on
the large map, and a corresponding sequence b1, b2, b3, ... on the small map.
It is not hard to show that the sequences converge to a point (city) that
is a fixed point of the mapping. As long as the scales differ, the fixed point
is unique.

--
Dave Seaman
ags@seaman.cc.purdue.edu

In article <19692.2b617217@ecs.umass.edu> padmanab@ecs.umass.edu writes:
>Suppose you have two maps of USA of different 
>scales. Now you place the smaller map inside 
>the bigger map such that it falls completely 
>within it. 
>
>Condider each point on the map to be a distinct 
>city!! 
>
>QUESTION: Are there any city/cities on the 
>smaller map which will coincide with the 
>city/cities  on the bigger map? If so why? 

Let's not be taken by the words "USA" and "city",
as some people got worried about.

The fixed point theorem says that every contracting projection
has a fixed point (i.e. f(x)=x), so that's it (need a proof? :-)

A simple way to see this in our present case (not a rigorous proof) is:
First suppose both maps are on the XY plane with same orientation,
i.e. no rotation in the XY plane is applied.
Then move the smaller map along the Z axis in parallel.
If we connect the corresponding points in both maps, the connecting
lines will converge at a single point.  Among those lines, there will
be one which is the shortest, which happens to be perpendicular
to the XY plane (i.e. the corresponding points have the same (x,y) coordinate).

For the general case, note that any positioning of the small map
can be obtained by transposition, and rotation around a single point.
Take this point to be the fixed point above (this can be alwas done
in combination with the proper transposition).
Applying rotation will leave the fixed point intact, though none of the
others will become another fixed point.
Incidentally, this shows that there is one and only one "city" that
coincides.

-Yuzuru Hiraga