# 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

${s}_{r}\left(\begin{array}{c}\hfill x\hfill \\ \hfill y\hfill \end{array}\right)=\left(\begin{array}{cc}\hfill 1\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill r\hfill \end{array}\right)\left(\begin{array}{c}\hfill x\hfill \\ \hfill y\hfill \end{array}\right)=\left(\begin{array}{c}\hfill x\hfill \\ \hfill ry\hfill \end{array}\right).$

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

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 |