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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 <math>X<math> and <math>Y<math> be metric spaces with metrics <math>|**|_X<math> and <math>|**|_Y<math> , a map <math>f:X\to Y<math> is called distance preserving if for any <math>x,y\in X<math> we have <math>|f(x) f(y)|_Y=|x y|_X.<math> 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 R<math>\to<math>R defined by
- <math> x\mapsto |x|<math>
is a path isometry but not a global isometry.
Metric spaces X and Y are called isometric if there is an isometry <math>X\to Y<math>. The set of isometries from a metric space to itself form a group with respect to compositon (called isometry group).
Examples
- 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 <math>f:X\to Y<math> between metric spaces such that for any point in the target space there is a point in the image on distance <math>\le\epsilon<math> and for any <math>x,y\in X<math> we have
- <math>\left||f(x)f(y)|-|xy|\right|\le\epsilon.<math>
- Note that ε-isometry is not assumed to be continuous.
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.
ja:等長写像 pl:izometria
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