Chapter 2
Inversions through a Circle
On her journey through our text Alice sees different models
of geometries. Chapter 5 presents her with a model of non-euclidean geometry
in which the "lines" are arcs of circles orthogonal to a given circle and
in chapter 3 Dr. Whatif describes a model of ordinary euclidean geometry
in which "lines" become circles! Some of the results
developed here in chapter 2 will be used in those models.
This chapter begins by motivating the definition of the
inversion (think 'reflection') of a point w.r.t. a circle. After the definition
is given, the chapter describes the inverted image of several common geometric
objects.
We begin with the notion of projecting the points of a
sphere onto a plane.
Definition: Stereographic Projection
Given a plane (imagine a horizontal plane), place a sphere
on the plane so that it rests on (is tangent at) a point, S, which we will
call the South Pole. Draw the diameter of the sphere that
passes through S and call the opposite endpoint, N, the North Pole.
A stereographic projection is the function whose domain is
all the points, P, on the sphere except the North Pole and whose range
is all the points of the plane, P'. To find P' when given P, draw
a line from the north pole through P and let P' be the intersection of
this line with the horizontal plane. One can similarly find P when
given P'.
Definition: Equator, etc.
The equator is the great circle on the sphere
formed by a horizontal plane through the center of the sphere.
Lattitude and longitude can be similarly
defined by analogy with those lines on the earth and we will omit the formal
statements.
Theorem: Equator Projection
The image of the equator under the stereographic projection
is a circle centered at the South Pole whose radius is twice that of the
equator.
Theorem: Symmetrical Point Projection Formula
Let Q1 and Q2 be points on the
same longitude of the sphere and symmetrical about the equator. Let the
stereographic projections of these points be P1 and P2
respectively. If S is the South Pole of the sphere and T is the image of
the point on the equator halfway between Q1 and Q2,
then ST2=SP1* SP2.
Since Q1 and Q2 are reflections on
the sphere through the equator and P1 and P2 are
images of Q1 and Q2, we will consider P1
and P2 to be 'reflections' through the circle that is the image
of the equator.The Symetrical Point Projection Formula gives a condition
for two points to be considered 'reflections' with respect to a circle,
that only depends on the center and radius of the circle. Thus the notion
of 'reflection' through a circle can be extended to any circle, not just
images of the equator. We state this formally in the next definition.
Definition: Inverse Image
Let K be a circle with center S. Let P1 be
any point in the plane except S. Let T be the point on the circle, K, intercepted
by the ray from S through P1 and define P2 to be
the point on the line through S and P1 such that ST2=SP1*SP2.
Then P1and P2 are called inverse images of
each other with respect to the circle K.
Note:
If P1 is on the circle K, then SP1=ST
so SP2=ST and P1=P2. So the inverse image
of any point on the circle of inversion is the point itself. In other words,
an inversion leaves every point on the circle of inversion fixed.
The next result describes the image of a circle when inverted
through another circle. While it is not surprising that the image a circle
reflected in a mirror is again a circle, it is surprising that the inverse
of a circle is also a circle in spite of the distortions of the circular
inversion map. The proof of the next theorem explains why we still get
a circle.
Theorem: Circle Inversion
Let K be a circle with center S and radius k. Let C1
be another circle not passing through S. Then the inverse image of the
points of C1 form another circle, C2, which is the
image of C1 under the dialation about point S with factor (k2)/S(C1).
Something special happens when the circle being inverted
is orthogonal to the circular "mirror":
Theorem: Orthogonal Circle Inversion
If K and C are orthogonal circles, then the image of
C is unchanged when inverted through circle K.
Proof:
There is one other situation where the circle does not change
when inverted.
Theorem: Coinciding Circles
If circles K and C coincide, the image of C when inverted
through K is again C.
The next theorem describes the image of a line when inverted
through a circle - provided the line misses the center of the "mirror"
circle.
Theorem: Line Inversion
Let K be a circle with center S. Let C be a circle passing
through S and SL1 the diameter of C passing through S. Let L2
be
the inverse of L1 w.r.t. circle K. Then the inverse of C w.r.t.
circle K is the line through L2 perpendicular to SL1.
Conversely, given a straight line, l, which does not pass through S, let
L2 be the point where the perpendicular from S meets l, Then
the inverse of l w.r.t. K is a circle which contains S and whose diameter
through S terminates at point L1where L1
is the inverse
of L2.
Proof:
It turns out that angles do not change when "reflected"
through a circle. We first need a definition of the angle between two circles
or a line and a circle.
Definition: Angle of intersection of two circles, etc.
The angle of intersection of two circles is the
angle between their tangents at the point of intersection. Similarly the
angle
of intersection of a line and a circle is the angle between the line
and the tangent to the circle at the point of intersection.
Theorem: Angle inversion
Many of our previous results are clearly related, although
some involved circles and others involved lines. Often the case with a
line seems to be the limiting case of the behvior of a case with the a
circle that gets very large.
We can take a more universal view and tie several results
together in one short statement with the following definition and theorem.
Definition: Extended plane, extended circles, etc.
The extended plane, E, is defined by E={p| p is
a usual point in the plane or p=<infinity>.}
An extended circle in E is either a usual circle
in the plane or a set of the form: l union <infinity>, where l is a
usual line in the plane.
Theorem: Extended Circle Inversion
The inverse of an extended circle with respect to another
extended circle is an extended circle.