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

22

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.

Get *Mathematical Basics of Motion and Deformation in Computer Graphics* now with O’Reilly online learning.

O’Reilly members experience live online training, plus books, videos, and digital content from 200+ publishers.