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).
Z 1
Z 2 W
Z
T
Figure 8.1.
Translations have the following properties:-(i)If Z 1 ≡(x 1 ,y 1 )Z 2 ≡(x 2 ,y 2 )Z≡(x,y),W≡(u,v),thentZ 1 ,Z 2 (Z)=W if and
only if
u=x+x 2 −x 1 ,v=y+y 2 −y 1.(ii) In all cases|tZ 1 ,Z 2 (Z 3 ),tZ 1 ,Z 2 (Z 4 )|=|Z 3 ,Z 4 |, so that each translation preserves
all distances.(iii) For each W∈Πthe equation tZ 1 ,Z 2 (Z)=W has a solution in Z, so that each
translation is an onto function.(iv) Each translation tZ 1 ,Z 2 has an inverse function tZ− 11 ,Z 2 =tZ 2 ,Z 1.(v)The translation tZ 1 ,Z 1 is the identity function onΠ.Geometry with Trigonometry
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