🚀 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. 
>
>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