Here, we introduce a quantity which is invariant under the Mobius
group, and "almost" invariant under the full inversive group. This
quantity is very significant geometrically, and admits several
interesting interpretations.
Definition
If u,v,w,z are distinct points of the complex plane,
then the inversive cross-ratio (u,v,w,z) is defined
by (u-w)(v-z)/(v-w)(u-z).
We can allow one of the variables to be ∞ by simply omitting
the corresponding factors. Thus, for example, if we take z = ∞,
(u,v,w,∞) = (u-w)/(v-w). This is equivalent to using the limit of
the general form as z tends to infinity.
If t is an inversive transformation, then we may apply t to each
variable. We shall write t(u,v,w,z) for the inversive cross-ratio
of the images, (t(u),t(v),t(w),t(z)).
A straight-forward calculation gives
Theorem II1
The inversive cross-ratio is invariant under the Mobius group M(2).
An indirect inversive transformation t maps z to m(z)*, where m
is a Mobius transformation. From Theorem II1, we can see that
| t(u,v,w,z) |
= (m(u)*,m(v)*,m(w)*,m(z)*) |
|
|
= (m(u),m(v),m(w),m(z))* |
|
|
= (u,v,w,z)* |
by Theorem II1 |
Convention We shall take the argument of a complex number
as the value in [-π,π].
Theorem II1*
If t is an inversive transformation, then it
(1) preserves the modulus of inversive cross-ratios, and
(2) preserves or reverses the argument of inversive cross-ratios.
As a first application, we can use the inversive cross-ratio give an
alternative approach to the invariants of similarity geometry.
Some further results, giving new proofs of familiar theorems, and
shedding new light on Ptolemy's Theorem are given in i-lines
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