Rotation

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Rotation

  1. 1. ROTATION
  2. 2. 1. ROTATIONJ FORMULA OVER O(0,0) . Let A(x,y) any point in plane V ang A’(x’,y’)isimage 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 
  3. 3. A’(x’,y’) A(x,y)  (0,0)
  4. 4. .• y’ = OA’ sin (+)• = OA(sin  cos  + cos  sin )• = x sin  + y cos • so• x’ = xcos  - y sin • y’ = x sin  + y cos • or  x cos   sin    x    y  sin   y    cos    
  5. 5. C’(x’,y’)=C’(x*’,y*’) y* C(x,y)=C(x*,y*) (a,b) x*(0,0) X Y
  6. 6. 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 :  x*  cos   sin    x *   y* sin     cos    y *  
  7. 7. In coordinate of X , Y axis , we have : xa  cos   sin    x  a   y b sin  cos    y  b     x cos   sin    x   p   y  sin  cos    y  q         p  a cos   b sin   a q  a sin   b cos   b
  8. 8. 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 :  SAB ,if s//tM t Ms    R P,θ , if t and s not paralel
  9. 9. A” t T A’ Q sP A
  10. 10. Theorem 1  R R P, P, - Theorem R P, R P,   R P,  
  11. 11. •If s perpendicular to t and P=(s,t) ,•then MtMs=HP. s A” P Et A D A’
  12. 12. . For every line a,b with• Teorema a//b, then MbMa=SCD with |CD|=2 x distance (a,b) and CD a. P’ P P’’ B A D a P b
  13. 13. Mb Ma = Mb I Ma = Mb (MsMs )Ma = Mb MsMs Ma = (Mb Ms )(Ms Ma) = HBHA = SCD with |CD|= 2 |AB|
  14. 14. •Translation SAB can represent ascomposition of two reflection Ms dan Mtwith s//t and s  AB, and distance of(s,t) is ½ |AB|. t B s A
  15. 15. • 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
  16. 16. Contoh permasalahan a A b B c C
  17. 17. • 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 ?
  18. 18. • 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 ?

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