21st CENTURY

FIGURE FIGURE
Mapping Surfaces onto Surfaces
gents. Like the external point of similarity, it also lies on the
axis connecting the center of the two circles. We can use
the internal point of similarity to map one circle onto the
other using the same principles as in Figures 1-3, but with a
slightly different result. By using the internal point of similarity,
we produce a "mirror" mapping of the circles, where
the left half of circle M is mapped onto the right half of the
larger circle M', and vice versa.
To map point P on circle M, draw through it an arbitrary
secant BC, which intersects with the circumference of circle
M at two points [Figure 6(b)]. Then project these points 6
and C through the internal point of similarity to find corresponding
points B' and C on the circumference of circle
M', and draw the secant there that corresponds to BC and
intersects the circumference of M'at points B' and C. Finally,
project point P through the internal point of similarity
to intersect with the secant of the larger circle B'C, determining
point P'.
As in the previous constructions, this demonstration also
provides a rigorous definition of the notion of parallel lines
as well as equal angles and similar triangles. Also, as in
Figures 1-3, the mapping produces a conformal mapping of
circle M onto circle M'.
Two circles need not lie outside each other, as they do in
Figures 1-3, to form the basis for establishing a mapping of
a surface onto itself. A rigorous mapping can be defined
just as well if the circles intersect or lie inside of each other.
There are four distinct cases, shown in Figure 7: (a) two
concentric circles; (b) two circles whose centers are eccen- Constructing I he Internal Point of Similarity
21 st CENTURY November-December 1988 55