Euclidean Motions

Abstract

In this chapter we study distance preserving maps, that is, the Euclidean motions. Since there are fewer Euclidean motions than affine maps, the classification is simpler. We also introduce a natural equivalence relation among Euclidean motions, similar to that for affine maps, and we characterize each equivalence class by a sequence of numbers (the coefficients of a polynomial and a metric invariant).

We associate a vector, the glide vector, to each Euclidean motion f. This vector, and in particular its module τ(f), plays an important role in the study and classification of Euclidean motions. In fact we have that

$$\tau(f)=\inf_{P\in \mathbb{A}} d(P,f(P)).$$

The subsections are

• 6.1 Introduction

• 6.2 Definition of Euclidean motion

• 6.3 Examples of Euclidean motions

• 6.4 Similar Euclidean motions

• 6.5 Calculations in coordinates

• 6.6 Glide vector

• 6.7 Classification of Euclidean motions

• 6.8 Invariance of the glide module

•  

  Exercises