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Many of the observations made about the group of the line can also be applied to *I*(**R**^{2}): the isometries of the plane **R**^{2}.

Again we can divide the isometries into two subsets: those for which the orthogonal transformation is in *SO*(2) (with determinant +1) and those with determinant -1. These are called **direct** and **opposite** symmetries respectively.

**Direct symmetries**Here is a complete classification of direct symmetries.

**Theorem**

*Any direct symmetry is either a translation or rotation about some point.***Proof**

- If
*f*has a fixed point**b**then*T*_{-b}*f**T*_{b}(**0**) =**0**and so this is a length preserving map which fixes the origin and hence is a linear map. Thus it is in*SO*(2) and is rotation about**0**. Hence*f*is rotation about the point**b**. - We have
*f*=*T*_{a}*L*with*L*∈*SO*(2) ⇒*f*(**x**) =**a**+*L*(**x**). Now we try to solve*f*(**x**) =**x**=**a**+*L*(**x**) to find a fixed point. That is, we look for a solution of (*L*-*I*)**x**= -**a**. The only case in which we could not find such a solution is if*L*-*I*were not invertible. That is, if +1 were an eigenvalue of*L*. But the only element in*SO*(2) with +1 as an eigenvalue is the identity. Hence if*f*fails to have a fixed point,*L*=*I*and*f*is a translation.

- If
**Opposite symmetries**We now classify the opposite symmetries. There are two kinds.

**Theorem**

*An opposite symmetry of***R**^{2}*is either reflection in any line or is a glide reflection.***Proof**

Note that the reflection may be in a line not necessarily through**0**.

A*glide reflection*(or*glide*) is a reflection in a line followed by a translation in a direction parallel to that line.Since any element in

*O*(2) -*SO*(2) is a reflection in a line through**0**, the result follows from:**Lemma**

*Let R*_{b}be reflection in a line through**0***containing the vector***b**. Then T_{a}R_{b}is a reflection if**a**and**b**are perpendicular and a glide-reflection otherwise.**Proof**

From the diagram, if**a**and**b**are perpendicular, then*T*_{a}*R*_{b}is reflection in the line*l*parallel to**b**through the point**a**/2.

If**a**and**b**are not perpendicular, then points of*l*are mapped to other points of*l*and so*f*acts on*l*by translation. Points on one side of*l*are mapped to the other side and so we have a*glide**along**l*.

- Note that glide reflections are really the most general form of an opposite symmetry. A reflection could be considered as a glide where the translation happens to be trivial.
- Glide reflections have no fixed points and so we get the following summary about isometries of
**R**^{2}.**Direct****Opposite**

**Fixed point**Rotation Reflection

**No fixed point**Translation Glide

- Products of direct symmetries are direct. It is however, possible for the product of rotations to be a translation.
- Products of reflections are rotations if the "mirror lines" meet and translations if they are parallel.
- Any isometry of
**R**^{2}can be written as a product of at most three reflections in lines.

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