Isometry

In geometry and mathematical analysis, an isometry is a bijective distance-preserving mapping.

General definitions

The notion of isometry comes in two main flavors: global isometry and a weaker notion path isometry or arcwise isometry. Both are often called just isometry and you should guess from context which one is used.

Let $X$ and $Y$ be metric spaces with metrics $|**|_{X}$ and $|**|_{Y}$ , a map $f:X\to Y$ is called distance preserving if it for any $x,y\in X$ we have $|f(x)f(y)|_{Y}=|xy|_{X}.$ A distance preserving map is automatically injective.

A global isometry is a bijective distance preserving map. A path isometry or arcwise isometry is a map which preserve lengths of curves (not nesesury bijective).

As an example, the map R$\to$R defined by

$x\mapsto |x|$

is a path isometry but not a global isometry.

Metric spaces X and Y are called isometric if there is an isometry $X\to Y$. The set of isometries from a metric space to itself form a group with respect to compositon (called isometry group).

Examples

1. In Euclidean space with the usual distance function, the (global) isometries can be characterized: there are no more than the 'expected' examples generated by rotations, reflections and translations. To put this more accurately, the isometries form a group, that is the semidirect product of the orthogonal group and the group of translations. See Euclidean group.

Generalizations

• ε-isometry or almost isometry also called Hausdorff approximation, it is a map $f:X\to Y$ between metric spaces such that for any point in the target space there is a point in the image on distance $\leq \epsilon$ and for any $x,y\in X$ we have

$\left||f(x)f(y)|-|xy|\right|\leq \epsilon .$

Note that ε-isometry is not assumed to be continuous.

• Quasi-isometry is yet an other useful generalization.

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Isometric projection or isometric view is the name given to a type of technical drawing / projection used in fields such as Mechanical Engineering or Architecture that makes an object/ building visible from three planes/co-ordinates.