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Geometric examples of transformations

Some transformations arise from very natural and intuitive geometric operations.

Identity

The laziest plane transformation of all is the one that does nothing. This transformation is known as the identity function $I : \mathbb{R}^2 \rightarrow \mathbb{R}^2$. It sends every point to itself, $I(x,y) = (x,y)$.

Rotations

Given a point $P$ in the plane, and an angle $\phi$,² we can consider rotating points around $P$ by the angle $\phi$. Any point $X$ can be rotated in this way, giving rise to a plane transformation $\mathbb{R}^2 \rightarrow \mathbb{R}^2$.

Translations

Given a vector ${\bf v}$, we can consider translating points by ${\bf v}$. Any point $X$ can be moved along by the vector ${\bf v}$ to obtain a point $X'$, and in this way we obtain a plane transformation.

Projections

Given a line $L$ in the plane, we can consider projecting points onto $L$. From a point $X$, draw the line through $X$ perpendicular to $L$; the intersection of this line with $L$ is the projection $X'$ of $X$ onto $L$. Projecting all points of the plane onto $L$, we obtain a plane transformation.

Reflections

Take a line $L$ and think of it as a mirror; we can then reflect points in $L$. Given a point $X$ not on $L$, suppose you are standing at $X$; your reflection $X'$ in $L$ is where you appear to be when you look in the mirror. Reflecting each point in this way we obtain a plane transformation.

To find $X'$, we can draw the line perpendicular to $L$ through $X$; $X'$ is the point on this perpendicular, on the other side of $L$, at the same distance from $L$ as $X$.

Dilations

Given a line $L$ and a real number $k$, we can dilate points from $L$ by a factor $k$. From a point $X$ not on $L$, once more we draw the perpendicular from $X$ to $L$. The dilation $X'$ of $X$ from $L$ with factor $k$ is the point $X'$ on the perpendicular which is $k$ times as far away from $L$ as $X$ is. In other words, if $P$ is the foot of the perpendicular from $X$ to $L$, then $\overrightarrow{PX'} = k \; \overrightarrow{PX}$.

Again we have a plane transformation, which stretches the plane out from $L$ by a factor of $k$. Note that if $k$ is positive, then $X'$ lies on the same side of $L$ as $X$; if $k$ is negative, then $X'$ lies on the opposite side of $L$ from $X$.

²This is the Greek letter phi.

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