Geometry with Trigonometry

Sec. 8.3 Translation of frame of reference 127

H 4 ′.Thenif(x′,y′)are the Cartesian coordinates ofZ′relative to([O′,I′,[O′,J′),

whenZ∈H 3 we haveZ′∈H 3 ′and so

x′=|O′,πO′I′(Z′)|=|O′,U′|=|O,U|=x.

Similarly whenZ∈H 4 we haveZ′∈H 4 ′and so

x′=−|O′,πO′I′(Z′)|=−|O′,U′|=−|O,U|=x.

Thusx′=xin all cases, and by a similar argumenty′=y.

8.2.2 .....................................

If l=ml(|BAC),thensl([A,B)=[A,C and sl([A,C)=[A,B.

Proof. We provesl([A,B)=[A,C as the other then follows. AsA∈lwe have

sl(A)=Aand so by 8.2.1(iii)sl([A,B)=[A,D for some pointD.

Suppose first thatA,B,Care collinear. WhenC∈[A,B we havel=AB,andso

sl(P)=Pfor allP∈[A,B.As[A,B=[A,C the conclusion is then immediate. On

the other hand whenA∈[B,C]so that|BACis straight,lis the perpendicular toABat

A.ThenifA=mp(B,D)we havesl([A,B)=[A,D,and[A,D=[A,CasD∈[A,C.

Finally suppose thatA,B,Care non-collinear. Now takeD∈[A,Cso that|A,D|=

|A,B|.IfM=mp(B,D)by 4.1.1(iv) we have thatl=AMand assl(B)=Dthen

sl([A,B)=[A,D=[A,C.

8.3 Translationofframeofreference ...................

NOTATION. By using 8.2.1(iii), (vi) and (xi), we showed in 8.2.1(xii) that
for any frame of reference F =([O,I ,[O,J) and any isometry f, F′=
([f(O),f(I),[f(O),f(J))is also a frame of reference, and that Cartesian coordi-
nates ofZrelative toFare also Cartesian coordinates of f(Z)relative toF′.We
denoteF′byf(F).
For any frame of referenceF=([O,I,[O,J),letZ 0 ≡F(x 0 ,y 0 )andF′=
tO,Z 0 (F).ThenifZ≡F(x,y)we have Z≡F′(x−x 0 ,y−y 0 ).
Proof. By 8.2.1(xii), tO,Z 0 (Z)
has coordinates(x,y)relative to
F′, and by 8.1.1(i) it also has
coordinate(x+x 0 ,y+y 0 )rela-
tive toF. Thus for all(x,y)∈
R×R the point with coordi-
nates(x+x 0 ,y+y 0 )relative to
F has coordinates(x,y) rela-
tive toF′. On replacing(x,y)by
(x−x 0 ,y−y 0 ), we conclude that
the point with coordinates(x,y)
relative toFhas coordinates
(x−x 0 ,y−y 0 )relative toF′.

O

O′

I

J I′

J′

H 1

H 2

H 4 H 3

Z

Z′

U

U′

V

V′

Figure 8.5.