|
A line L divides C+ into two disjoint regions, the sides of L. Here, ∞ is on the
extended line, so belongs to neither region.
A circle L divides C+ into two disjoint regions, the inside and the outside of L.
It is natural to regard ∞ as belonging to the outside. We shall refer to these
regions
as the sides of L.
It is natural to ask what effect inversive transformations have on the sides of an i-line.
the interior-exterior theorem
If L is an i-line and i an inversive transformation, then
i maps the sides of L to the sides of i(L).
This will follow quickly from an algebraic description of the sides, given in the
characterization lemma
If L is the i-line with equation f(z) = azz* -αz* - α*z + b = 0, then
(1) for any z ε C, f(z) is real, and
(2) the sides of L are the regions {z : f(z) > 0} and {z : f(z) < 0}.
(3) if L is a circle, the exterior is the set for which f(z) and a have the same sign.
proofs
If we invert a circle in a line, then it is clear that the interior inverts to the interior.
If we invert a circle in another circle, then the interior may map to the exterior.
Example 2
If we invert the circle L in the circle U : |z| = 1, then the interior of L
maps to the interior of the image if and only if 0 lies outside L.
Solution
Although we can do this by algebra, it is easier to think geometrically.
By the interior-exterior theorem, the interior of L maps to the interior or
to the exterior of the image.
The interior maps to the interior if and only if the exterior maps to the exterior.
Now 0 maps to ∞, and ∞ is in the exterior of the image L. Thus, the exterior maps
to the exterior if and only if 0 is outside L.
|