Theorem 2.1. E2 = T2
×fO2, where
f is the homomorphism O2 ®
Aut T2 given by the conjugation f(x)
= yxy-1, y Î
O2, x Î T2.
This means that we can identify the structure of E2
with the set of ordered pairs (t,s),
t Î T2
and
s Î O2,
where the product ×f is the
semidirect product. So, to every e Î
E2 corresponds an ordered pair from T2
×f O2:
In fact, for the product we have:
e1e2
= t1s1t2s2
= (t1s1t2s2s-1)(ss1)
= (t1f(s1)(t2))(s1s2)
® (t1,s1)(t2,s2). |
|
To simplify the manipulation with the transformations from E2,
we will introduce the following (analytic) notation. If
e
= ts, t((0,0)) =
v and matrix M Î O2
represents s in the standard base of R2,
we have that
t acts on the point x Î
R2 in the following way:
Now the bijection t « (v,M)
is evident. New structure containing the ordered pairs of elements of
R2 and O2 is analogous to the structure
of T2×f
O2 from the Theorem 2.1 (where R2 is
isomorphic to T2).
In this representation, transformations from E2 are
direct if det(M) = 1. Otherwise, they are indirect.
The isometries are represented as follows:
(a) Translation by a vector v as (v, I), where
I is the unit 2×2 matrix;
(b) Rotation anti-clockwise, through the angle q
about x, as (x-xM t,M), where
(c) Reflection in a line p as (2a,N), where the
image of p derived by the translation a is the line p¢
that contains the origin,
and y is the slope of p;
(d) Glide reflection by a vector b, in a line that translated
by b contains the origin, is represented as (2a+b,N),
with N defined as in (c).