## Links forward

### Transformations of the plane

#### Composition of transformations

The geometrical transformations of the plane we have discussed in this module are functions. For any of these transformations \(T\), we have

\[ T \colon \mathbb{R}^2 \to \mathbb{R}^2. \]For example, the translation 3 units to the right and 2 units down is defined by

\[ T_1(x,y) = (x+3,y-2). \]The stretch from the \(y\)-axis by a factor of 3 is the transformation defined by

\[ T_2 (x,y) = (3x,y). \]The reflection in the \(x\)-axis is given by

\[ R_x(x,y) = (x,-y). \]We now consider compositions of these transformations. For example, we have

\[ T_1\circ T_2(x,y) = T_1(T_2(x,y)) = T_1(3x,y) = (3x+3,y-2). \]Taking them in the opposite order, we obtain

\[ T_2\circ T_1(x,y) = T_2(T_1(x,y)) = T_2(x+3,y-2) = (3x+9,y-2). \]As expected, we did not get the same result. In general, composition of transformations of the plane is not commutative.

On the other hand, composition of translations is commutative. This is easily shown, as follows. Let \(S\) and \(T\) be two translations, defined by

\[ S(x,y) = (x+a,y+b) \qquad\text{and}\qquad T(x,y) = (x+c,y+d). \]Then

\[ S \circ T(x,y) = S(T(x,y)) = S(x+c,y+d) = (x+c+a,y+d+b) \]and

\[ T \circ S(x,y) = T(S(x,y)) = T(x+a,y+b) = (x+a+c,y+b+d). \]Hence, \(S \circ T = T \circ S\). (Note that two transformations of the plane \(T_1\) and \(T_2\) are equal if \(T_1(x,y) = T_2(x,y)\), for all \((x,y)\in \mathbb{R}^2\).)

#### Inverses of transformations

Again, consider the translation 3 units to the right and 2 units down, which is defined by

\[ T_1(x,y) = (x+3,y-2). \]We will find the inverse transformation. To do this, assume that there is a transformation \(T_4\), with \(T_4(x,y) = (x',y')\), such that \(T_1 \circ T_4(x,y) = (x,y)\). We have

\[ T_1(T_4(x,y)) = T_1(x',y') = (x'+3,y'-2), \]and therefore \(x'+3 = x\) and \(y'-2 = y\). This gives

\[ T_4(x,y) = (x',y') = (x-3,y+2). \]We can now check that \(T_4 \circ T_1(x,y) = (x,y)\) and \(T_1 \circ T_4(x,y) = (x,y)\), for all \((x,y) \in \mathbb{R}^2\). So \(T_4\) is the inverse of the transformation \(T_1\). That is, \(T_4 = T_1^{-1}\).

Next consider the reflection in the \(x\)-axis, given by

\[ R_x(x,y) = (x,-y). \]We note that \(R_x\circ R_x(x,y) = R_x(x,-y) = (x,-(-y)) = (x,y)\). This means that \(R_x\) is the inverse of itself. Earlier, we called this an involution.

#### Groups of isometries

Recall that an isometry is a distance-preserving map from the plane \(\mathbb{R}^2\) to itself.

Just as we have the identity function \(\mathrm{id}\) on \(\mathbb{R}\), we have the identity transformation \(I\) on \(\mathbb{R}^2\), given by

\[ I(x,y) = (x,y). \]This transformation is clearly an isometry.

Now let \(E\) be the set of all isometries of the plane. We make the following observations:

- •Closure.
- For all \(S, T \in E\), we have \(S\circ T \in E\).
- •Associativity.
- For all \(R, S, T\in E\), we have \((R\circ S)\circ T = R\circ(S\circ T)\).
- •Identity.
- There is \(I \in E\) such that, for all \(T \in E\), we have \(T\circ I = I\circ T = T\).
- •Inverses.
- For each \(T \in E\), there is an element \(T^{-1}\in E\) such that \(T \circ T^{-1} = T^{-1}\circ T = I\).

These are the four defining properties of a **group**. Note again that the commutative property does not necessarily hold.

Groups were first studied in depth by Galois when considering solutions of equations. Group theory provides a powerful tool for studying symmetries, and so groups are used in areas such as physics, chemistry and encryption.