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 $C$ :

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 $n=\operatorname{dim}(V)$ .

Title	strain transformation
Canonical name	StrainTransformation
Date of creation	2013-03-22 17:25:45
Last modified on	2013-03-22 17:25:45
Owner	CWoo (3771)
Last modified by	CWoo (3771)
Numerical id	5
Author	CWoo (3771)
Entry type	Definition
Classification	msc 15A04
Synonym	strain
Defines	strain coefficient