strain transformation

Let $E$ be a Euclidean plane. Fix a line $\mathrm{\ell }$ in $E$ and a real number $r\ne 0$. Take any point $p\in E$. Drop a line $m_p$ from $p$ perpendicular to $\mathrm{\ell }$. Denote $d(p,\mathrm{\ell })$ the distance from $p$ to $\mathrm{\ell }$. Then there is a unique point $p^{\prime }$ on $m_p$ such that

$$d(p^{\prime },\mathrm{\ell })=r\cdot d(p,\mathrm{\ell }).$$

The function $s_r:E\to E$ such that $s_r(p)=p^{\prime }$ is called a strain transformation, or simply a strain.

One can visualize a strain stretches a geometric figure if $|r|>1$ and compresses it if . If $r=1$, then $s_r$ is the identity function, the only time when a strain is a rigid motion. For example, let $\mathrm{\ell }$ be the $x$-axis and $C$ be a circle in the upper half plane of the $x$-$y$ plane. Then the following diagrams show how a strain transforms $C$:

Again, if $\mathrm{\ell }$ is the $x$-axis, then $s_r$ is the function that sends $(x,y)$ to $(x,ry)$. Representing the ordered pairs as column vectors and $s_r$ as a matrix , we have

Nevertheless, a strain, as a (non-singular) linear transformation, takes lines to lines, and parallel lines to parallel lines.

In general, given any finite dimensional vector space $V$ over a field $k$, a strain $s_r$ is a non-singular diagonalizable linear transformation on $V$ such that $s_r$ leaves a subspace $W$ of codimension $1$ fixed. $0\ne r\in k$ is called the strain coefficient.

Remark. By choosing an appropriate base for $V$ of dimension $n$, $s_r$ can be represented as a diagonal matrix whose diagonals are $1$ in at least $n-1$ cells and $r$ in at most one cell.

It is easy to see that every non-singular diagonalizable linear transformation on $V$ can be written as a product of $n$ strains, where $n=\dim (V)$.