Definition:Isometry (Euclidean Geometry)

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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)