Isometry Guide, Meaning , Facts, Information and Description

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

Table of contents

1 General definitions 1.1 Examples 2 Generalizations

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 and be metric spaces with metrics and , a map is called distance preserving if for any we have 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 preserves the lengths of curves (not necessarily bijective).

As an example, the map RR defined by

is a path isometry but not a global isometry.

Metric spaces X and Y are called isometric if there is an isometry . 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 between metric spaces such that for any point in the target space there is a point in the image on distance and for any we have

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.

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