8 Translations; axial symmetries; isometries
COMMENT. In this chapter we introduce translations and develop them and axial
symmetries. These will be useful in later chapters. It is more convenient to frame our
proofs for isometries generally.
8.1 Translations and axial symmetries .................
8.1.1 .....................................
Definition. Given points
Z 1 ,Z 2 ∈Π,we define atrans-
lation tZ 1 ,Z 2 to be a function
tZ 1 ,Z 2 :Π→Π such that, for
allZ∈Π,tZ 1 ,Z 2 (Z)=Wwhere
mp(Z 1 ,W)=mp(Z 2 ,Z).