“Given a metric space[1] (loosely, a set and a scheme for assigning distances between elements of the set), an isometry is a transformation which maps elements to the same or another metric space such that the distance between the… elements in the new metric space is equal to the distance between the elements in the original metric space.”
For example, “any reflection, translation and rotation is… [an] isometry on Euclidean spaces.[2]“
Notes:
- Isometries are also called a congruence, or a congruent transformation
- “An isometry is automatically injective[3]“
- Isometries are “usually assumed to be bijective.[3]”
- If an isometry is bijective, it can also be called a “global isometry, isometric isomorphism or congruence mapping.” Reflection, translations, and rotations are global isometries on Euclidean spaces.
- “Two metric spaces X and Y are called isometric if there is a bijective isometry from X to Y.”
- “An isometry is an isomorphism[4] of metric spaces.”*
Source of quotes/information: “Isometry,” Wikipedia, retrieved 11/7/2020, emphasis added to first paragraph
* The source of this line is different from the rest: “Isomorphism,” Wikipedia, retrieved 11/7/2020.
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- [1] What is the Gist of a “Metric Space” or a “Metric” (Mathematics)?
- [2] What is the Gist of “Euclidean Space”?
- [3] What is the Gist of “Injective,” “Surjective,” and “Bijective” Functions (Mathematics)?
- [4] What is the Gist of an “Isomorphism”?
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Disclaimer:
I am not a professional in this field, nor do I claim to know all of the jargon that is typically used in this field. I am not summarizing my sources; I simply read from a variety of websites until I feel like I understand enough about a topic to move on to what I actually wanted to learn. If I am inaccurate in what I say or you know a better, simpler way to explain a concept, I would be happy to hear from you :).
