Plane Transformations

A plane tranformation is a rule which assigns to every point P in the plane another point P' in the plane. In math lingo, we often denote a tranformation as

<math> f: \R^{2} \longrightarrow \R^{2} </math>.

Examples

Let's write some examples of plane transformations in this notation.

• Translation by <math> \vec c = \left[\begin{array}{c}a\\b\end{array}\right]</math>

<math> f(x, y) = (x + a, y+b) </math>.

• Reflection about the line <math> x= y</math>

<math> f(x,y) = (y,x) </math>

• Rotation about the origin by <math>\theta </math> degrees

<math> f(x,y) = (x\cos{\theta} - y\sin{\theta}, x\cos{\theta} + y\sin{\theta} </math>

• Scaling with center at origin and factor k

<math> f(x.y) = (kx, ky) </math>

Connection with Matrix Multiplication

Let's try to write the examples above in vector notation.

• Translation can be written as <math> f (\vec v) = \vec v + \vec c </math>.

To write the other examples above in vector notation, we will need to use matrix multiplication.

Given a matrix <math> A = \left[\begin{array}{rr} a & b \\ c & d \\ \end{array}\right] </math> and a vector <math> \vec v = \left[\begin{array}{c}x\\y\end{array}\right]</math> , we can define their product as

<math> A \vec v = \left[\begin{array}{c}ax + by\\cx + dy\end{array}\right]</math>

Each of the remaining examples of tranasformations can be expressed as a matrix multiplied by a vector; when this is true, we say that the transformation is represented by a matrix .

The matrix for reflection about <math> x= y </math> is given by <math> \left[\begin{array}{rr} 0 & 1 \\ 1 & 0 \\ \end{array}\right] </math>

The matrix for rotation bout the origin by <math> \theta </math> is <math> \left[\begin{array}{rr} \cos{\theta} & -\sin{\theta} \\ \sin{\theta} & \cos{\theta} \\ \end{array}\right] </math>

The matrix for scaling with center at the origin and factor k is <math> \left[\begin{array}{rr} k & 0 \\ 0 & k \\ \end{array}\right] </math>