Content

Transformations of the plane

A plane transformation is just a function from the plane to itself, i.e. a function $F: \mathbb{R}^2 \rightarrow \mathbb{R}^2$. Such an $F$ takes a point $(x,y)$ to another point, which we can denote $(x',y')$ or $F(x,y)$.¹ You can think of $x,y$ as inputs to $F$, and $x',y'$ as outputs of $F$.

By our definition, any function $\mathbb{R}^2 \rightarrow \mathbb{R}^2$ whatsoever is a plane transformation. E.g.:

\[ F(x,y) = (0,5), \quad F(x,y) = (2x+y, 5x-3y), \quad F(x,y) = (x-2y+3, -4x+5y-6), \quad \] \[ F(x,y) = (x^2 + y^3, e^y \sqrt[3]{x} ), \quad F(x,y) = ( \lfloor xy \rfloor, \sin (x+y) ). \]

(However, in this module we will not deal with functions as complicated as the last two.)

It's not possible to draw a graph of a function $F : \mathbb{R}^2 \rightarrow \mathbb{R}^2$ in the same way we draw the graph $y = f(x)$ of a function $f: \mathbb{R} \rightarrow \mathbb{R}$. For a function $f: \mathbb{R} \rightarrow \mathbb{R}$, its graph consists of a plot of all points $(x,y)$ such that $y = f(x)$. To draw a similar graph of a function $F: \mathbb{R}^2 \rightarrow \mathbb{R}^2$, we would need to plot all $(x,y,x',y')$ such that $(x',y') = F(x,y)$, which would require four coordinates — it would have to be drawn in four-dimensional space! As we will see, however, there are other ways to visualise a plane transformation.

In this module, we will do the following.

• We will briefly examine some geometry, introducing a few examples of plane transformations arising from geometric operations.
• Then, we will examine some algebra of vectors and matrices, and discuss linear transformations, which arise from this algebra.
• Next, we will examine some geometric transformations in depth: rotations, translations, projections, dilations, and more. We will investigate the relationship between the geometry and algebra of these operations.
• Finally, we will discuss transformations of graphs, discussing what happens to a graph $y=f(x)$ when it is subject to a transformation.
• In the Links Forward section, we will discuss some further concepts such as determinants, isometries, and how the ideas of this module can be generalised from the 2-dimensional plane to 3-dimensional space, and even to higher dimensions.

Warning regarding notation. Throughout this module, we often blur the distinction between a point $(x,y)$ in the plane, and its position vector $\begin{bmatrix} x \\ y \end{bmatrix}$. This is a slight abuse of notation, but it may be useful, and we hope it will not be too confusing.

¹Note the primes on $x',y'$ have nothing to do with derivatives!

Next page - Content - Geometric examples of transformations