# Difference between revisions of "Isometry"

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− | An '''isometry''' is a map which preserves | + | An '''isometry''' is a map which preserves [[distance]]s between [[point]]s. Isometries exist in any space in which a distance function is defined, i.e. an arbitrary abstract [[metric space]]. In the particular case where we take our space to be the usual Euclidean plane or Euclidean 3-space (<math>\mathbb{R}^2</math> or <math>\mathbb{R}^3</math> with the standard [[Euclidean metric]]), the isometries are known as ''rigid motions'' and two sets which can be transformed onto each other by an isometry are said to be [[congruent (geometry) | congruent]]. |

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+ | ==Isometries are injective== | ||

+ | Since for any metric we have <math>d(x, y) > 0</math> whenever <math>x \neq y</math>, it follows that every isometry must be an [[injection]]. | ||

+ | ===Proof=== | ||

+ | Suppose otherwise. Then there is some non-injective isometry <math>\phi: S \to T</math>. Since <math>\phi</math> is not injective, we know <math>\exists x, y \in S</math> such that <math>x \neq y</math> and <math>\phi(x) = \phi(y)</math>. But <math>x \neq y \Longrightarrow d_S(x, y) > 0</math> while <math>\phi(x) = \phi(y) \Longrightarrow d_T(\phi(x), \phi(y)) = 0</math>, and this contradicts the fact that <math>\phi</math> is an isometry. | ||

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+ | Note that this does ''not'' mean that isometries are necessarily [[bijection]]s. Consider, for example, the [[discrete metric]] on the integers, <math>(\mathbb{Z}, d)</math> such that <math>d(x, x) = 0</math> and <math>x\neq y \Longrightarrow d(x, y) = 1</math>. It is simple to verify that this is a metric space. The map <math>\phi: \mathbb{Z} \to \mathbb{Z}</math> given by <math>\phi(n) = 2n</math> is an isometry but it is not [[surjection | surjective]]. | ||

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## Latest revision as of 17:17, 23 September 2006

An **isometry** is a map which preserves distances between points. Isometries exist in any space in which a distance function is defined, i.e. an arbitrary abstract metric space. In the particular case where we take our space to be the usual Euclidean plane or Euclidean 3-space ( or with the standard Euclidean metric), the isometries are known as *rigid motions* and two sets which can be transformed onto each other by an isometry are said to be congruent.

## Isometries are injective

Since for any metric we have whenever , it follows that every isometry must be an injection.

### Proof

Suppose otherwise. Then there is some non-injective isometry . Since is not injective, we know such that and . But while , and this contradicts the fact that is an isometry.

Note that this does *not* mean that isometries are necessarily bijections. Consider, for example, the discrete metric on the integers, such that and . It is simple to verify that this is a metric space. The map given by is an isometry but it is not surjective.

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