Definition:Isometry (Euclidean Geometry)
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This page needs the help of a knowledgeable authority. In particular: I am trying to strike a balance between the mathematically rigorous definition of an isometry based on the context of a metric space (see Definition:Isometry (Metric Spaces) and an intuitive understanding of such a mapping as one is taught at school, with that intuitive idea of "the plane" and "ordinary space", so as to bridge the gap between rotations/reflection/translation etc. in space and the general metric space with its general definition of a distance, without burdening the reader with the concept of a general metric. Any inspiration welcome. If you are knowledgeable in this area, then you can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by resolving the issues. To discuss this page in more detail, feel free to use the talk page. When this work has been completed, you may remove this instance of {{Help}} from the code.If you would welcome a second opinion as to whether your work is correct, add a call to {{Proofread}} the page. |
This page is about isometry in the context of Euclidean geometry. For other uses, see isometry.
Definition
Let $\EE$ be a real Euclidean space.
Let $\phi: \EE \to \EE$ be a bijection such that:
- $\forall P, Q \in \EE: PQ = P'Q'$
where:
- $P$ and $Q$ are arbitrary points in $\EE$
- $P'$ and $Q'$ are the images of $P$ and $Q$ respectively
- $PQ$ and $P'Q'$ denote the lengths of the straight line segments $PQ$ and $P'Q'$ respectively.
Then $\phi$ is an isometry.
That is, an isometry is a bijection which preserves distance between points.
Context
An isometry is defined usually for either:
- $n = 2$, representing the plane
or:
- $n = 3$, representing ordinary space.
Also known as
An isometry is also known as an isometric mapping, or an isometric map.
Texts which approach the subject from the direction of applied mathematics and physics refer to an isometry as a rigid motion.
Also see
- Results about isometries in the context of Euclidean geometry can be found here.
Sources
- 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): isometry (isometric map)
- 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): isometry (isometric map)