2. Euclidean GroupIn the R^{2}-model of Euclidean plane, isometries (distance preserving transformations) are represented as functions f : R^{2} ® R^{2} satisfying the following condition:
The set of all isometries is a group with the composition of transformations as the binary operation. After decomposing the Euclidean group E_{2} into it's characteristic subgroups, we could easily describe the properties of E_{2}. 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 T_{2} is the translational subgroup T_{2} of E_{2}, T_{2} < E_{2}. (3) The set of rotations with the origin as a center, and reflections in lines containing the origin, represents the subgroup O_{2} of E_{2}, O_{2} < E_{2}. This is the orthogonal subgroup of the Euclidean group, denoted as O_{2}. So, every element e Î E_{2} can be represented as:
We see that O_{2} ÇT_{2} = e, where e is the identity transformation. Hence, every element of E_{2} we can decompose in the product of elements of O_{2} and T_{2} in a unique way (otherwise, e = st = s^{¢}t^{¢} Þ t^{-1} t^{¢} = s^{-1}s^{¢} and as O_{2} ÇT_{2} = 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 T_{2} is a normal subgroup of E_{2}. As sts^{-1} Î T_{2} and O_{2} ÈT_{2} generate E_{2}, T_{2} is the normal subgroup of E_{2} (for the proof see [3]). From that and (4) it follows:
In fact, for the product we have:
To simplify the manipulation with the transformations from E_{2}, we will introduce the following (analytic) notation. If e = ts, t((0,0)) = v and matrix M Î O_{2} represents s in the standard base of R^{2}, we have that t acts on the point x Î R^{2} in the following way:
Now the bijection t « (v,M) is evident. New structure containing the ordered pairs of elements of R^{2} and O_{2} is analogous to the structure of T_{2}×_{f} O_{2} from the Theorem 2.1 (where R^{2} is isomorphic to T_{2}). In this representation, transformations from E_{2} 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,
(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). |