 2.4. REFLECTIONS IN 2D 9
is bears a resemblance to (2.8), but the role of rotations and translations are reversed.
2.4 REFLECTIONS IN 2D
A reﬂection (ﬂip) with respect to a line y D .tan /x through the origin can be expressed as
R
1 0
0 1
R
D
cos 2 sin 2
sin 2 cos 2
: (2.10)
A reﬂection is orientation-reversing transformation which preserves the shape. e determinant
of the matrix (2.10) is 1. e composition of two reﬂections is a rotation, and the resulting
rotation depends on the order of compositions of reﬂections:
cos 2 sin 2
sin 2 cos 2
cos 2
0
sin 2
0
sin 2
0
cos 2
0
D R
22
0
: (2.11)
e totality of rotations and reﬂections form an orthogonal group of size two, which is de-
ﬁned by
O.2/ D fA 2 M.2; R/ j AA
T
D I
2
g: (2.12)
e totality of rotations has been denoted by
SO.2/ D fA 2 O.2/ j det.A/ D 1g: (2.13)
Figure 2.4: Connected components.
In the terminology in Section 3.2, SO.2/ is a normal subgroup of O.2/. Any reﬂection
is not considered to be a motion, since it cannot be continuously connected with the identity
transformation. In other words, O.2/ is not connected while SO.2/ is connected. Note that a
connected component is like an island, illustrated in Figure 2.4. With this terminology, SO.2/ is a
connected component. It is known that any two 2D reﬂections are continuously connected. is
means that the set of 2D reﬂections is a diﬀerent connected component of O.2/ from SO.2/.
e same holds for an arbitrary dimension n. at is, SO.n/ is connected, and O.n/ has two
connected components. e set of the elements of O.n/ whose determinants are 1 is the other
connected component of SO.n/. Moreover, for any two elements g; h 2 O.n/ with g; h SO.n/,
we have gh 2 SO.n/. is fact is rephrased as the index of the subgroup SO.n/ in O.n/ is two,
and denoted by ŒO.n/ W SO.n/ D 2.

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