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Homestrain transformation

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# strain transformation

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

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

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 $|r|<1$. If $r=1$, then $s_{r}$ is the identity function, the only time when a strain is a rigid motion. For example, let $\ell$ 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 $C$:

unit=1.5cm \pspicture(-4,-2)(5,3) \psline(-4,0)(4,0) \rput(4.5,0)$\ell$ \psellipse(-3,1)(0.5,0.5) \psellipse(-1,2)(0.5,1) \psellipse(1,0.5)(0.5,0.25) \psellipse(3,-1)(0.5,0.5) \rput(-3,-2)$C$ \rput(-1,-2)$s_{2}(C)$ \rput(1,-2)$s_{{\frac{1}{2}}}(C)$ \rput(3,-2)$s_{{-1}}(C)$

Again, if $\ell$ 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

$s_{r}\begin{pmatrix}x\\ y\end{pmatrix}=\begin{pmatrix}1&0\\ 0&r\end{pmatrix}\begin{pmatrix}x\\ y\end{pmatrix}=\begin{pmatrix}x\\ ry\end{pmatrix}.$

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. $0\neq r\in k$ 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 $n=\operatorname{dim}(V)$.

## Mathematics Subject Classification

15A04*no label found*

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