ROTATION

1. ROTATIONJ FORMULA OVER O(0,0) .

Let A(x,y) any point in plane V ang A’(x’,y’)is

image of A over R ,0 , or A’ = R ,0 (A).

Let m(XOA)= .

We have x =OA cos dan y = OA sin and

x’ = OA’ cos (+)

= OA (cos cos - sin sin)

= x cos - y sin

A’(x’,y’)

A(x,y)

(0,0)

.• y’ = OA’ sin (+)• = OA(sin cos + cos sin )• = x sin + y cos • so• x’ = xcos - y sin • y’ = x sin + y cos • or

y

x

y

x

cossin

sincos

'

'

(0,0)

(a,b)C(x,y)=C(x*,y*)

C’(x’,y’)=C’(x*’,y*’)

Y

X

x*

y*

2. ROTATION OVER P(a,b)

• Let we have coordinate system with centre P(a,b)and has two axis X* and Y*, X//X* and Y//Y*.

• If C(x*,y*) and C’=RP,(C), then C’ (x*’,y*’) , we have a relation :

*

*

cossin

sincos

*'

*'

y

x

y

x

In coordinate of X , Y axis , we have :

by

ax

by

ax

cossin

sincos

'

'

q

p

y

x

y

x

cossin

sincos

'

'

bbaq

abap

cossin

sincos

THEOREM• Rotation RP, can represent in composition of two lines reflection over s and t with P is (s,t) and m(<(s,t))=½ .

• Rotation is an isometry

• composition of two lines reflection :

paralelnot s and t if ,R

s//t ,ifSMsMt

θP,

AB

A” t T A’ Q s P A

Theorem

RP,RP,RP,

R - P,R 1P,

Theorem

s

A”

P E

t

A D A’

•If s perpendicular to t and P=(s,t) , •then MtMs=HP.

.

• Teorema For every line a,b with a//b, then MbMa=SCD with |CD|=2 x distance (a,b) and CD a.

abP

B

P’

A

P

D

P’’

Mb Ma = Mb I Ma

= Mb (MsMs )Ma

= Mb MsMs Ma

= (Mb Ms )(Ms Ma)

= HBHA

= SCD with | CD| = 2 | AB|

•Translation SAB can represent as composition of two reflection Ms dan Mt with s//t and s AB, and distance of (s,t) is ½ |AB|.

A

Bs

t

• Given three paralel lines a, b dan c. • Construct an equilateral triangle

ABC with condition A on a, B on b and C on c.

a

b

c

Contoh permasalahan

a

b

c

A

B

C

• a. Fixed any point A on a.• b. Rotated line c, with angle 60o over A, we got c’.• c. Intersection of line c’ and line b, ( c’,b) is point

B .• d. We can construct equilateral triangle ABC.•  • We can also start with fixed point B on b or C on

c.

• Can do it ?

• a. Fixed any point B on b.• b. Rotated line a, with angle 60o over B, we got a’.• c. Intersect of line a’ and line c, ( a’,c) is point C .• d. We can construct equilqteral triangle ABC.•  • We can also start with fixed point A on a or C on

c.

• Can do it ?