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2. Euclidean Group




In the R2-model of Euclidean plane, isometries (distance preserving transformations) are represented as functions f : R2 ® R2 satisfying the following condition:

|f(x)-f(y)| = |x-y|,     x,y Î R2.

The set of all isometries is a group with the composition of transformations as the binary operation. After decomposing the Euclidean group E2 into it's characteristic subgroups, we could easily describe the properties of E2.

We will now mention some of the properties of isometries [1, 2, 6, 7, 8, 9], relevant for the problem considered:

(1) There are the four types of isometries: translations and rotations (direct isometries), reflections and glide reflections (indirect isometries);

(2) The set of all translations T2 is the translational subgroup T2 of E2, T2 < E2.

(3) The set of rotations with the origin as a center, and reflections in lines containing the origin, represents the subgroup O2 of E2, O2 < E2. This is the orthogonal subgroup of the Euclidean group, denoted as O2.

(4) Every element of E2 can be represented as a composition of one rotation with the origin as a center or one reflection in a line passing through the origin, and one translation.

So, every element e Î E2 can be represented as:

e = st,
where s Î O2 and t Î T2. From this relationship follows that E2 is the semidirect product of it's subgroups O2 and T2.

We see that O2 ÇT2 = e, where e is the identity transformation. Hence, every element of E2 we can decompose in the product of elements of O2 and T2 in a unique way (otherwise,

e = st = s¢t¢ Þ t-1 t¢  = s-1s¢

and as O2 ÇT2 = e Þ t = t¢ and  s = s¢ , so we have the contradiction). For satisfying the conditions of the Theorem A.1.1, it is necessary to show that T2 is a normal subgroup of E2. As sts-1 Î T2 and O2 ÈT2 generate E2, T2 is the normal subgroup of  E2 (for the proof see [3]).

From that and (4) it follows:

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:

(t,s) Û e = ts.

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:

e(x) = v+xM t 

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

M æ
ç
è
cosq -sinq
sinq cosq
ö
÷
ø

(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,

N æ
ç
è
cosy siny
siny -cosy
ö
÷
ø
,
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).

 

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