The Euclidean Group

Abstract

The isometries of the plane form a group under composition of functions, the so called Euclidean group E2. A function g: R² ¡ª> R² belongs to E2, provided it preserves distance; that is to say

$$||g\left( x \right) - g\left( y \right)|| = ||x - y||$$

(1)

for every pair of points x, y in ℝ². If g, h ∈ E2, we have

$$||g\left( {h\left( x \right)} \right) - g\left( {h\left( y \right)} \right)|| = ||h\left( x \right) - h\left( y \right)||$$

(2)

because g is an isometry

$$= ||x - y||$$

(3)

because h is an isometry; therefore, gh ∈ E ₂. Composition of functions is associative, and the identity transformation of the plane acts as identity element Finally, each g ∈ E ₂ is a bijection and satisfies

$$||{y^{ - 1}}\left( x \right) - {g^{ - 1}}\left( y \right)|| = ||g\left( {{g^{ - 1}}\left( x \right)} \right) - g\left( {{g^{ - 1}}\left( y \right)} \right)||$$

(4)

because g is an isometry

$$= ||x - y||$$

(5)

so g⁻¹ ∈ E ₂ and we do indeed have a group.