21st CENTURY
21st CENTURY
21st CENTURY
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FIGURE<br />
(a)<br />
(b)<br />
Projective Relations Among 3 Circles<br />
62 November-December 1988 <strong>21st</strong> <strong>CENTURY</strong><br />
FIGURE<br />
17<br />
Constructing Tangent Circles<br />
similarity for two pairings of the three circles. Notice that<br />
these two internal points of similarity lie on a straight line<br />
with one of the external points of similarity constructed<br />
from another pairing of the circles.<br />
Steiner's Influence on Cantor<br />
The line connecting the three external points of similarity<br />
in Figure 16(a) is called a "line of threefold similarity." By<br />
means of it, the three circles form a closed projective system.<br />
Map a point P from circle M to circle M' and find its<br />
image P' with respect to external point of similarity A3; then<br />
map that image point P' onto the third circle M" and determine<br />
its image point P" by means of external point of similarity<br />
A,. Finally, map that image point P " back to the first<br />
circle from which we started, circle M. We find that we end<br />
up at the same point P from which we started the series of<br />
projections. This illustrates that the space determined by<br />
three circles (or spheres) or any number of circles (or<br />
spheres), projectively related in this way, is closed—something<br />
upon reflection we would expect from the similarity<br />
of all circles to each other.
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