If L is a circle with centre C, then we can extend the domain to include C by introducing
a new
"point" ∞, and setting iL(C) = ∞, iL(∞) = C.
If L is a line, iL is defined for z ≠ ∞, and we set iL(∞) = ∞.
Now an inversion i in L is defined on C+ = Cu{∞}.
It is easy to check that i2 is the identity, so i is a transformation of C+.
Also, if i(P) = Q, then i(Q) = P, so we say that {P,Q} are inverse with respect to L.
We noted earlier that, for z ≠ ∞, iL(z) = z if and only if z ε L.
If L is a circle, then iL(∞) ≠ ∞.
If L is a line, then iL(∞) = ∞, so we should include ∞ as a point of the "line".
We say that L+ = Lu{∞} is an extended line.
Definition A subset of C+ is an i-line if it is a circle or an extended line.
Note that an i-line L is an extended line if and only if ∞ ε L.
Unfortunately, the set of inversions is not closed under composition.
For example, we have met the inversions i(z) = z* and j(z) = 1/z*
The composite is ioj(z) = 1/z, not an inversion since inversions involve z*.
To get a group of transformations,
we make the
Definition
The inversive group I(2) is the group generated by inversions in i-lines.
Its elements are inversive transformations.
the inversion theorem
If {P,Q} are inverse with respect to an i-line L and i is an inversive transformation, then
(1) i(L) is an i-line, and
(2) {i(P),i(Q)} are inverse with respect to i(L).
proof
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