rotation
TRANSCRIPT
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ROTATION
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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
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A’(x’,y’)
A(x,y)
(0,0)
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.• 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
'
'
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(0,0)
(a,b)C(x,y)=C(x*,y*)
C’(x’,y’)=C’(x*’,y*’)
Y
X
x*
y*
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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
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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
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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
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A” t T A’ Q s P A
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Theorem
RP,RP,RP,
R - P,R 1P,
Theorem
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s
A”
P E
t
A D A’
•If s perpendicular to t and P=(s,t) , •then MtMs=HP.
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.
• 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’’
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Mb Ma = Mb I Ma
= Mb (MsMs )Ma
= Mb MsMs Ma
= (Mb Ms )(Ms Ma)
= HBHA
= SCD with | CD| = 2 | AB|
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•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
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• 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
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Contoh permasalahan
a
b
c
A
B
C
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• 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 ?
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• 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 ?