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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 X and Y be metric spaces with metrics | * * | X and | * * | Y , a map is called distance preserving if it for any 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 R 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
- 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
-
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- 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.
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