Sec. 8.2 Isometries 123
and thus
|sl(Z 3 ),sl(Z 4 )|^2 =1
(a^2 +b^2 )^2{
[(b^2 −a^2 )(x 3 −x 4 )− 2 ab(y 3 −y 4 )]^2+[− 2 ab(x 3 −x 4 )−(b^2 −a^2 )(y 3 −y 4 )]^2}
=
1
(a^2 +b^2 )^2{
[(b^2 −a^2 )^2 + 4 a^2 b^2 ][(x 3 −x 4 )^2 +(y 3 −y 4 )^2 ]+[− 4 ab+ 4 ab](b^2 −a^2 )(x 3 −x 4 )(y 3 −y 4 )}
=(x 3 −x 4 )^2 +(y 3 −y 4 )^2.(ii) For ifmis the line throughZwhich is perpendicular tolandW=sl(Z),then
W∈mand soπl(W)=πl(Z).Thenπl(W)=mp(W,Z)soZ=sl(W). This shows that
the functionslis its own inverse.
8.2 Isometries ............................
8.2.1 .....................................
DefinitionA functionf:Π→Πwhich satisfies|Z 1 ,Z 2 |=|f(Z 1 ),f(Z 2 )|for all points
Z 1 ,Z 2 ∈Π, is called anisometryofΠ.
Each translation and each axial symmetry is an isometry.
Proof. This follows from 8.1.1.
Each isometry f has the following properties:-(i)The function f:Π→Πis one-one.(ii) For all Z 1 ,Z 2 ∈Π,f([Z 1 ,Z 2 ]) = [f(Z 1 ),f(Z 2 )], so that each segment is mapped
onto a segment, with the end-points corresponding.(iii) For all distinct points Z 1 ,Z 2 ∈Π, f([Z 1 ,Z 2 )=[f(Z 1 ),f(Z 2 ), so that each
half-line is mapped onto a half-line, with the initial points corresponding.(iv) For all distinct points Z 1 ,Z 2 ∈Π,f(Z 1 Z 2 )=f(Z 1 )f(Z 2 ), so that each line is
mapped onto a line. If f(Z)∈f(Z 1 )f(Z 2 )then Z∈Z 1 Z 2.(v)If Z 1 ,Z 2 ,Z 3 are noncollinear points, then f([Z 1 ,Z 2 ,Z 3 ])≡
[f(Z 1 ),f(Z 2 ),f(Z 3 )].(vi)Let Z 3 ∈l andH 1 ,H 2 be the closed half-planes with common edge l, with
Z 3 ∈H 1 .LetH 3 ,H 4 be the closed half-planes with common edge f(l), with
f(Z 3 )∈H 3 .Thenf(H 1 )⊂H 3 ,f(H 2 )⊂H 4.(vii) The function f:Π→Πis onto.(viii) In(vi),f(H 1 )=H 3 ,f(H 2 )=H 4.