>Exactly how is the given circle identified without
>identifying a center point or a point whose distance from
>the center point represents the circle radius?

A circle may have been drawn using a stencil, or in the usual manner - by somebody else's anscetor - so long ago that the paper, like an old map, grew so decayed and fragile that while the lucky adventurer tried to smoothen it out on a table, the middle part caved in and disintegrated.

Have you seen a metaphorical interpretation of Bottema's theorem:

http://www.cut-the-knot.org/Curriculum/Geometry/Bottema.shtml

>This makes
>little sense either theoretically or mechanically.

Why, it did a lot of sense to several generations of geometers starting in the mid 1800s.

>At least, my compass can not do it.

It certainly takes imagination rather than brute force.

>Maybe there needs to be a
>new postulate to Euclid's elements that every circle has a
>center point.

By all means. You should write to somebody responsible for improving on Euclid's axioms.

>Does not the "transformation" of the circle (in three
>dimensions) represent a "redrawing" of the circle;

I do not know. It all might be in the mind's eye. How do you draw a circle in 3d?