21st CENTURY

FIGURE 12
(a) (b)
Mapping Corresponding Points Using Corresponding Radii
through a discontinuity. We now have the following situation:
(1) Straight lines in either circle (except for radii) correspond
to curves in the other circle.
(2) There seems to be no way to construct secants that
correspond to each other between the two circles. A secant
through the finite circle M would correspond to a curve
connecting two points of the infinite circumference of infinite
circle M', but there seems to be no way to determine
the path that that curve would take.
(3) Thus we must map corresponding points by using
corresponding radii or arcs or some other method besides
mapping corresponding secants. Radii intersecting the circle
at only one point can still be found that correspond
between the two circles. But the third sign that we have
passed through a discontinuity is that corresponding radii
are not parallel, as they are in mappings between finite
circles. All the radii of the infinite circle radiate from its
infinitely distant center. As a result, all are perpendicular to
the straight line that forms the visible portion of its circumference
and intersects the axis connecting the circles at
point A' (Figure 11). On the other hand, the circumference
of the finite circle is not straight; its radii are therefore not
parallel to each other.
(4) Last, but not least, only half of the points on the axis
connecting the infinite and finite circles can be mapped
onto each other.
First, how do we now map points within and on the circles
so that they correspond? Figure 12(a) shows how to map
points of finite circles onto each other through the internal
point of simihrity, by means of constructing the corresponding
radii on which they lie. We will use this technique
to map the inf nite circle onto the finite. To map point P of
circle M into circle M' in Figure 12(a), draw the radius MB
on which it lie:, then project the point 6 at which that radius
intersects the :ircumference, through the internal point of
similarity to fir d on the circumference of circle M' the point
B' to which B c orresponds. Then draw the radius that intersects
B', and d aw aline from the poi nt P in ci rcle M th rough
the internal p )int of similarity to intersect radius M'B' in
circle M' at th< point P' that corresponds to P.
We will use this method to map the infinite circle onto
the finite circ e, keeping in mind that in this case corresponding
radi will not be parallel. Figure 12(b) shows how
to map a point P' in the infinite circle onto its corresponding
point P in fini e circle M. Figure 13(a) shows this method
applied to pro ect triangle P'R'T from the infinite circle into
the finite circli i.
The first thir g to notice is that the points that correspond
to points P', C ', R' and P', S', T, which lie on straight lines
in the infinite :ircle, do not lie on straight lines in the finite
circle; they lie on curved arcs. This reflects the discontinuity
in passing f om the finite to the infinite.
The mapping procedure in Figure 13 is identical to that
used in Figure 6(b) with the exception that corresponding
radii are not parallel. Points P', Q' and R' in the infinite
circle are conr ected to its circumference via perpendicular
lines, segmen s of the infinite radii on which they lie. The
points where t lese radii intersect the circumference X', Y',
T are then pro jected through the internal point of similarity
21st CENTJRY November-December 1988 59